the hilbert transform of schwartz distributions and applications

THE HILBERT TRANSFORM OF SCHWARTZ DISTRIBUTIONS AND APPLICATIONS

PURE AND APPLIED MATHEMATICS

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HARRY HOCHSTADT, PETER LAX, JOHN TOLAND A complete list of the titles in this series appears at the end of this volume.

THE HILBERT TRANSFORM OF SCHWARTZ DISTRIBUTIONS AND APPLICATIONS

J. N. PANDEY Carleton University

A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York · Chichester · Brisbane · Toronto · Singapore

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Library of Congress Cataloging in Publication Data: Pandey, J. N.

The Hilbert transform of Schwartz distributions and applications / by J. N. Pandey.

p. cm. — (Pure and applied mathematics) Includes bibliographical references. ISBN 0-471-03373-1 (cloth : alk. paper) 1. Hilbert transform. 2. Schwartz distributions. I. Title.

II. Series: Pure and applied mathematics (John Wiley & Sons : Unnumbered) QA432.P335 1996 515'.782—dc20 95-18944

10 9 8 7 6 5 4 3 2 1

To my parents (Pandit Chandrika Pandey and Shrimati Chameli Devi), as well as to all of the 329 passengers and crew members

of Air India flight number 182 which crashed on June 23,1985 near the Irish coast.

CONTENTS

Preface

1. Some Background

1.1. Fourier Transforms and the Theory of Distributions, 1 1.2. Fourier Transforms of L2 Functions, 4

1.2.1. Fourier Transforms of Some Well-known Functions, 4 1.3. Convolution of Functions, 7

1.3.1. Differentiation of the Fourier Transform, 12 1.4. Theory of Distributions, 12

1.4.1. Topological Vector Spaces, 13 1.4.2. Locally Convex Spaces, 21 1.4.3. Schwartz Testing Function Space:

Its Topology and Distributions, 23 1.4.4. The Calculus of Distribution, 29 1.4.5. Distributional Differentiation, 31

1.5. Primitive of Distributions, 31 1.6. Characterization of Distributions of Compact Supports, 32 1.7. Convolution of Distributions, 33 1.8. The Direct Product of Distributions, 34 1.9. The Convolution of Functions, 36 1.10. Regularization of Distributions, 39 1.11. The Continuity of the Convolution Process, 39 1.12. Fourier Transforms and Tempered Distributions, 40

1.12.1. The Testing Function Space 5(K"), 40 1.13. The Space of Distributions of Slow Growth S'(W), 41

viü CONTENTS

1.14. A Boundedness Property of Distributions of Slow Growth and Its Structure Formula, 41

1.15. A Characterization Formula for Tempered Distributions, 42 1.16. Fourier Transform of Tempered Distributions, 44 1.17. Fourier Transform of Distributions in D'(W), 49

Exercises, 51

2. The Riemann-Hilbert Problem 54

2.1. Some Corollaries on Cauchy Integrals, 54 2.2. Riemann's Problem, 56

2.2.1. The Hubert Problem, 58 2.2.2. Riemann-Hilbert Problem, 58

2.3. Carleman's Approach to Solving the Riemann-Hilbert Problem, 58

2.4. The Hubert Inversion Formula for Periodic Functions, 66 2.5. The Hubert Transform on the Real Line, 75 2.6. Finite Hubert Transform as Applied to Aerofoil Theories, 82 2.7. The Riemann-Hilbert Problem Applied to Crack Problems, 84 2.8. Reduction of a Griffith Crack Problem to the Hubert Problem, 85 2.9. Further Applications of the Hubert Transform, 86

2.9.1. The Hilbert Transform, 86 2.9.2. The Hibert Transform and the Dispersion Relations, 86 Exercises, 87

3. The Hilbert Transform of Distributions in DJ,, 1< p < » 89

3.1. Introduction, 89 3.2. Classical Hilbert Transform, 91 3.3. Schwartz Testing Function Space, Ό^, K p < °°, 93

3.3.1. The Topology on the Space Ό^, 93 3.4. The Hilbert Transform of Distributions in £>(,, K p < ° ° , 96

3.4.1. Regular Distribution in !){?, 96 3.5. The Inversion Theorem, 97

3.5.1. Some Examples and Applications, 98 3.6. Approximate Hilbert Transform of Distributions, 100

3.6.1. Analytic Representation, 103 3.6.2. Distributional Representation of Analytic Functions, 104

3.7. Existence and Uniqueness of the Solution to a Dirichlet Boundary-Value Problem, 107

3.8. The Hubert Problem for Distributions in V[p, 110 3.8.1. Description of the Problem, 110 3.8.2. The Hubert Problem in D[„ l</><°°, 111 Exercises, 113

The Hubert Transform of Schwartz Distributions

4.1. Introduction, 114 4.2. The Testing Function Space H(D) and Its Topology, 115 4.3. Generalized Hubert Transformation, 116 4.4. An Intrinsic Definition of the Space H(D) and Its Topology, 118 4.5. The Intrinsic Definition of the Space H(D), 121

4.5.1. The Intrinsic Definition of the Topology of H(D), 121 4.6. A Gel'fand-Shilov Technique for the Hubert Transform, 121

4.6.1. Gel'fand-Shilov Testing Function Spaced, 122 4.6.2. The Topology of the Space Φ, 124

4.7. An Extension of the Gel'fand-Shilov Technique for the Hubert Transform, 125 4.7.1. The Testing Function Space Si, 126 4.7.2. The Testing Function Space Z\, 126 4.7.3. The Hubert Transform of Ultradistributions in Z,', 127

4.8. Distributional Hubert Transforms in n-Dimensions, 131 4.8.1. The Testing Function Space Si (Kn), 131 4.8.2. The Testing Function Space ZiCR"), 131 4.8.3. The Testing Function Space S^R"), 132 4.8.4. The Testing Function Space ZV(W), 132 4.8.5. The Strict Inductive Limit Topology of Z^W), 132 Exercises, 136

R-Dimensional Hubert Transform

5.1. Generalized «-Dimensional Hubert Transform and Applications, 138 5.1.1. Notation and Preliminaries, 138 5.1.2. The Testing Function Space O^R"), 138 5.1.3. The Test Space X(W), 139

5.2. The Hilbert Transform of a Test Function in X(W), 142 5.2.1. The Hilbert Transform of Schwartz Distributions in

O]f(W),p> 1, 145 5.3. Some Examples, 147 5.4. Generalized (n + l)-Dimensional Dirichlet

Boundary-Value Problems, 149

X CONTENTS

5.5. The Hubert Transform of Distributions in £>^(R"), p > 1, Its Inversion and Applications, 151 5.5.1. The «-Dimensional Hubert Transform, 152 5.5.2. Schwartz Testing Functions Space Ό(Μ"), 152 5.5.3. The Inversion Formula, 153 5.5.4. The Topology on the Space Dy,(W), 154 5.5.5. The «-Dimensional Distributional Hilbert Transform, 156 5.5.6. Calculus on T>'U(W), 157 5.5.7. The Testing Function Space //(©(IR")), 159 5.5.8. The «-Dimensional Generalized Hubert Transform, 160 5.5.9. An Intrinsic Definition of the Space H(O(U"))

and Its Topology, 161 Exercises, 168

6. Further Applications of the Hubert Transform, the Hubert Problem—A Distributional Approach 170

6.1. Introduction, 170 6.2. The Hubert Problem, 174 6.3. The Fourier Transform and the Hubert Transform, 178 6.4. Definitions and Preliminaries, 180 6.5. The Action of the Fourier Transform on the Hubert Transform,

and Vice Versa, 181 6.6. Characterization of the Space F(S0(R")), 182 6.7. The p-Norm of the Truncated Hubert Transform, 183 6.8. Operators on LP(W) that Commute with Translations

and Dilatations, 186 6.9. Functions Whose Fourier Transforms Are Supported on Orthants, 192

6.9.1. The Schwartz Distribution Space ©^(R"), 194 6.9.2. An Approximate Hubert Transform and Its Limit in LP(U"), 197 6.9.3. Complex Hubert Transform, 202 6.9.4. Distributional Representation of Holomorphic Functions, 205 6.9.5. Action of the Fourier Transform on the Hubert Transform, 206

6.10. The Dirichlet Boundary-Value Problem, 213 6.11. Eigenvalues and Eigenfunctions of the Operator H, 214

Exercises, 215

7. Periodic Distributions, Their Hubert Transform and Applications 217

7.1. The Hubert Transform of Periodic Distributions, 217 7.1.1. Introduction, 217

CONTENTS χι

7.2. Definitions and Preliminaries, 222 7.2.1. Testing Function Space Pn, 111 122. The Space P2'T of Periodic Distributions, 223

7.3. Some Well-Known Operations on Ρ'2τ, 228 7.4. The Function Space L£T and Its Hilbert Transform, p ^ 1, 228 7.5. The Inversion Formula, 230 7.6. The Testing Function Space Q2T, 231 7.7. The Hilbert Transform of Locally Integrable and Periodic Function of

Period 2T, 233 7.8. Approximate Hilbert Transform of Periodic Distributions, 236

7.8.1. Introduction to Approximate Hilbert Transform, 237 7.8.2. Notation and Preliminaries, 238

7.9. A Structure Formula for Periodic Distributions, 241 7.9.1. Applications, 245 Exercises, 247

Bibliography 249

Subject Index 255

Notation Index 259

PREFACE

For the past two decades I have been researching the Hilbert transform of Schwartz distributions. I and my colleagues have arrived at many new results. These results form the basis of this book which will be of interest not only to mathematicians but also to engineers and applied scientists. My objective is to demonstrate the wide applicability of Hilbert transform techniques. This book may be used either as a graduate-level textbook on the Hilbert transform of Schwartz distributions and periodic distributions or as a research monograph.

The Hilbert transform (///)(*) = ±(P) /Γ«, ^¡dt arises in many fields such as

i. Signal processing (the Hilbert transform of periodic functions) ii. Metallurgy (Griffith crack problem and the theory of elasticity) iii. Dirichlet boundary value problems (potential theory) iv. Dispersion relation in high energy physics, spectroscopy, and wave equations v. Wing theory

vi. The Hilbert problem vii. Harmonic analysis

The Hilbert problem during the last four decades has received considerable atten-tion in metallurgical problems, namely in the Griffith crack problem in the theory of elasticity. Sneddon and Lowengrub who have been pioneers of applying the finite Hilbert transform in the theory of elasticity state: "The major development of the present century in the field of two-dimensional elasticity has been Muskhelishvili's work on the complex form of the two-dimensional equations due to G. B. Kolsov." Consequently a fair amount of treatment of classical as well as distributional Hilbert problems has been incorporated in the book. In particular. Chapter 2 is devoted to the classical Hilbert problem, whereas Chapter 3 and Chapter 6 cover distributional Hilbert problems.

The singular nature of the kernel '_() of the Hilbert transform has made the work on the Hilbert transform very difficult to accomplish and in turn the work on

xiii

XIV PREFACE

the Hubert transform of distributions has suffered. Nevertheless, the problem of the Hubert transform of distributions has received the attention of many mathematicians who had started working on the Hubert transform of various subspaces of Schwartz distributions. Among them are Laurent Schwartz [87], GePfand and Shilov [44], Horvath [5], Bremermann [9], Jones [53], Lauwerier [58], Tillmann [97,63], Beltrami and Wohlers [6], Orton [72], Mitrovic [61, 62, 63], and Carmichael [15, 16]. The approach to the Hubert transform of distributions that I have developed with my colleagues is the simplest and the most effective. It is easily accessible to applied scientists despite the fact that I have used a fairly advanced treatment in this book.

Among many new results that I wish to point out are the inversion formula for the n-dimensional Hubert transform H2f = (-\)"f, n > 1 and a new definition for the Hubert transform of periodic functions with period 2τ:

(Hf)(x)= -lim (P) f Άώ (i) ■n tf-.» J_N x - t

= bP)Lf{x-t)coiOdt (ii)

This identity is true at least for the class of functions / E Lplr My definition of the

Hubert transform of periodic functions is a generalization of the Hubert transform of periodic functions with period 27r, defined as

("/)(*) = T - C ) /" / ( * - ')cot l- dt (iii) 2.TT J_„ 2

Definition (iii) was widely used by Butzer, Nessel, Oppenheim, Schaefer, and many others, and to the best of my knowledge, there has been no formula or definition for the Hubert transform of periodic functions with period other than 2ττ. I also believe that the definition of the Hubert transform of periodic functions in the form (i) will be especially useful to people working in signal processing for computational purposes. From definition (i), which is the definition for the Hubert transform of functions, a unified theory of the Hubert transform of periodic as well as nonperiodic functions can be developed.

In Chapter 71 develop the theory of the Hubert transform of periodic distributions and also the approximate Hubert transform of periodic distributions. I use this theory to find a harmonic function U(x, v) which is periodic in x with period 2τ vanishes as y —► oo, uniformly V* E IR and tends to a periodic distribution / (with period 2τ) as y —* 0+ , in the weak distributional sense. The uniqueness of the solution is also proved.

My discussion proceeds from a Paley-Wiener type of theorem (Theorem 6.18) which gives the characterization of functions or generalized functions whose Fourier transform vanishes over certain orthants or the union of orthants of R".

In Chapter 5 I also give a generalization of the Hubert problem

F+(x) - F-(x) = f(x)

PREFACE xv

in higher dimensions and solve it. In Section 6.7 I calculate the p-norm \\H\\P of the Hubert transform operator H:L"(W) -> LP(U"), p > 1. In Theorem 6.3 I give a characterization of bounded linear operators on LP(W), p > 1, which commute with translation and dilatation.

Another highlight of the book is the very elegant treatment of the one-dimensional Hubert transform of distributions in D'u, p > 1, in Chapter 3. Chapter 3 will be especially useful to applied scientists.

The book assumes that the reader has a background in the elements of functional analysis. Chapter 1 essentially deals with the prerequisite materials for the theory of distributions and Fourier transform.

Chapter 2 presents the Riemann-Hilbert problem and gives the background mate-rial to the study of the Hubert transform. It includes sections on the appearance of the Hubert transform in wing theory, in the theory of elasticity, in spectroscopy, and in high-energy physics.

Chapter 3 discusses the Hubert transform of Schwartz distributions in D'u and related boundary value problems.

Chapter 4 considers the Hubert transform of Schwartz distributions in D'. It also discusses a Gel'fand and Shilov technique for the Hubert transform of generalized functions and an improvement to their techniques.

Chapter 5 deals with «-dimensional Hubert transform and the approximation technique in evaluating the Hubert transform and the inversion formulas. The Hubert transform of distribution in D¡j,(U") is also covered, and many applications are given.

Chapter 6 considers the applications of the Hubert transform to Riemann-Hilbert problems (classical as well as distributional). Many other related results are presented. One among many is the derivation of a Paley-Wiener theorem.

Chapter 7 deals with the periodic distributions and their Hubert transforms. With the firm belief that perfection never comes without practice I have included

numerous examples in every chapter. I wish to acknowledge the assistance of Professor E. L. Koh of the University of

Regina, and of Professor S. A. Naimpally and Dr. James Bondar of my department who were very kind and patient in going through various chapters of the manuscript and gave me very useful suggestions. I want to express my sincere gratitude to Professor Angelo Mingarelli of my department who very patiently entered the graphic designs on my manuscript and helped me consult CDRAM (Math Reviews) for the preparation of the manuscript.

I further wish to thank Mr. Andrew E. Dabrowski a former student at Carleton, Mr. Sanjay Varma of the Mehta Research Institute Allahabad India, Ms. Nalini Sreeshylan of the Institute of Sciences Bangalore India, Mr. K. P. Sivaraman of the Tata Institute of Fundamental Research, Bombay, Mrs. Shelly Bereznin, Mr. Ibrahim Farah, and Mrs. Diane Berezowski of Carleton University for their help in the typing of the manuscript in its various forms.

The major part of the typing was done by Mrs. Diane Berezowski who modified all the chapters typed by others and unified them into a single TgX scheme along with her own typing. She never lost her temper despite the many changes I had asked

xv¡ PREFACE

her to incorporate in the manuscript. I am grateful to her for her patience and for her active and prompt cooperation.

In addition I would like to thank Drs. Q. M. Tariq and Ehab Bassily of my department who very patiently prepared the Index and the Notation Index of this book.

I also wish to acknowledge a grant from the Natural Sciences and Engineering Research Council of Canada in support of my research. I wish to thank the Tata Insti-tute of Fundamental Research, Bombay, the Indian Institute of Science, Bangalore, and the Mehta Research Institute, Allahabad, for financially supporting my visits to their institutions, where a considerable part of the research work on this monograph was completed. In particular, I express my gratitude to Professor H. S. Mani, Director, Mehta Research Institute of Mathematics and Mathematical Physics for encouraging me to use the facilities of his institute.

My debt to my wife, Krishna for her constant support and cheerfulness under difficult circumstances when the manuscript was under preparation, is so profound as to defy description.

J. N. PANDEY

Ottawa, Canada

1 SOME BACKGROUND

1.1. FOURIER TRANSFORMS AND THE THEORY OF DISTRIBUTIONS

This chapter discusses some very important properties of the Fourier transform of functions that will be useful in developing the theory of the Fourier transform of distributions. It also develops some basic results concerning topological vector spaces, in particular, locally convex spaces, and extends these results to develop a theory of distributions and tempered distributions.

Definition. Let / be a function of a real variable t defined on the real line. Then its Fourier transform F{w) is defined by the relation

(J/)(w) = F(w) = [ f(t)eiw,dt (1.1)

provided that the integral exists.

There are many variations on definition (1.1). Some authors add the factor ¿ . o r

-ir= outside the integral sign, and some take the kernel of the Fourier transform as e~'M in place of the kernel e'w'. Some authors including L. Schwartz have written the kernel of the Fourier transform e2mwl. But these variations matter little.

The inverse Fourier transform of / in our case will be defined as

J"7(0 = T- I fMe~iw'dw (1.2)


Example 1. Let

'(Ήο - 1 < / < 1 elsewhere

The Hubert Transform of Schwartz Distributions and Applications by J. N. Pandey

Copyright © 1996 John Wiley & Sons, Inc.

2 SOME BACKGROUND

/

i C 2sinw

Then

r\ ( 2sinw w

2 when w = 0

Note that the function /(f) £ ¿ ' but (Jf)(w) g ΖΛ

Theorem 1. Let / G L1. Then

■ / i. F(w) = / f(t)e'w'dt is well defined V w E R.

ii. F(w) is uniformly continuous and bounded on U. iii. F(w) —> 0 as |w| —> oo.

Proo/. (i) /Γ«, |/(iy'""l Λ := /Γ«, 1/(01 Λ· Clearly /(/)e,M" is a measurable function of /. Therefore /(/)elwl is absolutely integrable, and it is integrable for each w G R. Hence F(w) is well defined for each w G R.

(ii) |F(w)| < / " „ |/(0e''""| A =s / Γ , Ι/(/)Ι A. F(w) is uniformly bounded. Now we prove that F(w) is uniformly continuous on /?. Choosing /V large enough so that for an arbitrary e > 0, we have

J \f(f)\dt + j_ |/(0lA<^ (1.3)

A simple calculation shows that

AF = F(w + Aw) - F(w) rN -I f(t)[ei(w+Aw)' - eiM]dt

N

-N (IM) + / + / ) f(t)e'w'le'*wl - 1]A (1.4)

Now denote the first integral in the right hand side of (1.4) by / and the second pair of integrals by 7:

Γ ι/ωι ι^ J-N

l/l* / 1/(011«*"*-U A

By virtue of the uniform continuity of (e'^wl — 1), we can choose δ small enough so that

|/| < | whenever |Δνν| < δ (1.5)

FOURIER TRANSFORMS AND THE THEORY OF DISTRIBUTIONS 3

δ being independent of / (and w as well).

|7| s (J+J W ( / ) | A < | (1.6) Combining (1.5) and (1.6), we have

|AF| ^ e whenever |Δνν| < δ

This proves the uniform continuity of F(w) over the real line. (iii) The space T)(W) of infinitely differentiable functions with compact support

on W is dense in LP(W) p > 1, and the identity map from D{W) to Z/(R") is continuous [67, 101]. Let now φ G ΤΗβ) be such that

J -o \f(t)-<p{t)\dt<€-.

Then

F(w)= f [f(t) - <p(t)]eiM dt + / <p(t)eiMdt (1.7)

Denoting the two integrals in the right hand side of (1.7) by J\ and J2, respectively, we see that

/

oo

\m - <p{t)\dt < ^ (1.8)

A simple integration by parts shows that

Ji —> 0 as \w\ —» oo

Therefore there exists a k > 0 such that

| / 2 l < | v M > * (1.9)

Combining (1.8) and (1.9), we get for c > 0, that there exists a constant k > 0 such that \F(w)\ < e V |»v| > k. Since e is arbitrary our result is proved. D

Theorem 2. Let / e O and F(w) be the Fourier transform of / . Assume that F(w) eZ-'.Then

1 Γ fit) = — / F(w)e-,w'dw a.e. (1.10) 2TT y_M

The equality (1.10) holds at all points of continuity of / . Proof of this inversion formula for the Fourier transform can be found in many books on integral transforms.

The Fourier transform of a function fit) defined from W —» U is defined as F(w) = J"R„ f(t)e"'w dt, provided that the integral exists. Here t = (t\,t2,...,t„) and

4 SOME BACKGROUND

w = (ΗΊ , H»2, . . . , w„), t ■ w = t\ w\ + t2w2 + · · · + t„w„. Theorems analogous to Theorems 1 and 2 are valid. The inversion formula analogous to (1.10) is

f(0= ( ¿ ) j Fiwye-^'dw a.e.

which is also valid if/ and F both £ L\W).

1.2. FOURIER TRANSFORMS OF L2 FUNCTIONS

1.2.1. Fourier Transforms of Some Well-known Functions

Consider

Tie

Jo

e-Meiw'dl

cos wte

w2 + 1

J[h(t- 1) - A(f - 2)] = i eiw'dt

where A(f) is Heaviside's unit function,

eiw, |2

iw |,

J

'e2'* - eiw' iw

1 1 Γ eilw J

= 7r[<T"A(w) + e+"'A(-H')].

a different category from the above is

J ( l ) = /" ί*·*(Λ 7-00

which does not exist in the classical sense but does exist in the distributional sense, as will be proved later.

We can verify that our inversion formula as stated before is valid in the case of these functions. For example,

rx[e~wh(w) + ewh(-w)]TT = - Í - / 2w J0

lre-»e-iwldw + -Í-7T / ewe-'w'dw 2π 7°

J - 0 0

FOURIER TRANSFORMS OF L1 FUNCTIONS 5

- i. Γ ] i ~ 2 [ 1 + it 1 -it _ 1 ~ TTT5

Theorem 3. Let /(f) be continuous, and let / ' ( /) be piecewise continuous on the real line such that lim|,|_oo /(f) = 0, and /(f) is Fourier transformable V w E l . Then /'(w) is also Fourier transformable V w G U, and

JX/'Xw) = ( -w)(J / ) (w)

Proof. Consider the operations

J ( / ' ) ( w ) = / fXtW^dt J-oo

= e'w7(f)| - / iweiw'f(t)dt I-» 7-»

= (-iw)f eiMf{t)dt J -00

= (-/w)J(/)(w)

These operations can be justified by integrating between - M and N and letting M,N -+O0. □

Corollary 1. If / ' is continuous on R and is Fourier transformable, and if lim|(|_oo/(f) = 0, then /(f) is also Fourier transformable and (,Tf')(w) = (-/w)(J/)(w).

Corollary 2. If /(n)(f) is Fourier transformable and is continuous such that \\m,^±xfk\t) = 0 for k = 0,1,2 n - 1, then f{n~x\ f-n~2),..., f, f are all Fourier transformable and

J(/(*>)(w) = (-/wO*(J/)(w), ¿ = 1 , 2 n

Corollary 3. If/ is continuously differentiable up to order n such that limi,^,» /(*'(f) = 0 for each k = 0 ,1,2, . . . , n - 1 and /(f) is Fourier transformable, then each of the derivatives / ' , / " /<,l) is Fourier transformable and

nf{k))(w) = (-i*0*(J/)(w). * = 1,2,3 «.

Corollary 4. I f / / ' , / " , . . . , / ( n _ 1 ) are all continuous and / ( n ) is piecewise con-tinuous in any aribtrary, finite closed interval of R and if limj,^» /(*'(f) = 0 for each k = 0,1,2 n — 1 and /(f) is Fourier transformable, then /(*'(f) is Fourier transformable and T(f{k))(w) = (-/'νν)*(^)(νν) for each k = 1,2,3 n.

6 SOME BACKGROUND

For functions defined over a finite measure space, every / £ L2(X) belongs to L\X), but this result is not true here in general. Let us consider

/w = s in /

t 1,

ίΦΟ

t = 0

This function / £ Ü (R), but it does belong to L2(R). There are functions that belong to L'(R) and L2(U) as well. For example e~M £ ¿'(R) Π L2(R).

The Fourier transform of functions belonging to L2(R) does not necessarily exist in general in the pointwise sense. Also, if / £ L2(R), then the truncated function f(t)xi-a*] -+ /(f) »n L2(R) as a -» ». Since f(t)x{-a,a] £ L'(R) Π L2(R), the space of functions belonging to ¿'(R) Π L2(R) forms a dense subset of L2(R). The question now arises as to how the Fourier transform of / £ L2(R) is to be defined.

Using the above-mentioned density property, Plancherel proved the following well-known theorem [3, p. 91], which is called the Plancheral theorem.

Theorem 4. Let / £ L2(R). Then there exists a function f(w) £ L2(R) such that

/(w) -r J -a

f(tVw'dt asa —+ o° (1.Π)

that is,

/(w) = l.i.m. / "-"" J-a

f(t)e'w'dt

is,

fix) - ¿ f h")e-iwxdw

f(x) = l.i.m. -i- Γ )

asa

{w)e',wxdw

(1.12)

For a measure theoretic and modern proof see Rudin [84] on the real and complex analysis.

It is further proved that

/(w) dw

Γ eiw' - 1 a.e. (1.13)

and

/(*) dx f - 1

—iw f(w) dw a.e. (1.14)

CONVOLUTION OF FUNCTIONS 7

A very elementary proof of the fact that (1.13) and (1.14) are eqivalent to (1.11) and (1.12), respectively, is given by Akhiezer and Glazman in their work on the theory of linear operators in Hilbert space [3, pp. 75-76]. These concepts of the Fourier transform were further developed by Titchmarch [99] for Lp functions 1 oo

as N —> oo.

The Fourier receprocity relation also holds in the sense that

d /(£) =

d

Γ ¿* - 1 / / w — — d t

J -00

™-iiS-m It

e '** -I

a.e.

-ι"ί άζ a.e.

Also

\\f\\q^K(p)U \f(x)\pdx\ 1/(P-I)

where K(p) is a constant depending upon p. Thus the Fourier integral operator J is a bounded linear operator from W to Lq. The work of generalizing the Fourier transform of functions was continued by Laurent Schwartz [87] who put forward the theory of the Fourier transform of tempered distributions and L. Ehrenpreis [36] who brought forward his theory of the Fourier transform of Schwartz distributions.

1.3. CONVOLUTION OF FUNCTIONS

Let / and g be complex-valued functions defined on the real line, which we denote by R. Then their convolution ( / * g)(x) is defined by

(/ * i)W -Í fix - y)g(y)dy

provided that the above integral exists. At the set of points where the convolution exists, we are able to define a new function ( / * g)(x). Since many of the properties of the convolution defined above are similar to the product, we call the convolution

8 SOME BACKGROUND

if * g)(x) a convolution product of two functions f and g. It is a simple exercise to show that ( / * g)(x) = ig * f)(x). I will now give some results in the form of theorems that demonstrate the existence of the convolution. The proof of the following theorem is found in Rudin [84, pp. 146-147].

Theorem 6. Let/ ,g e L'(-°°,°°). Then

Γ i· if * g)ix) = I fix ~ y)giy) dy exists and is finite a.e.

ii. The function ( / * g)ix) G V ( - » , oo). iü. I l / * i l l i s | | / l l i l l g l l . .

where

\dx = [ \m\t J -00

Proof. There is no loss of generality in assuming that / and g are Borel measurable. Clearly, if / and g are Lebesgue measurable, then there exist Borel measurable functions /o and go, respectively, defined on the real line such that f = fo a.e. and g = go a.e. Borel measurable functions are necessarily Lebesgue measurable, so we may assume that / and g are Borel measurable functions. Also the value of an integral remains unchanged by changing the values of the integrand at a set of points of measure zero. Now define

Fix.y) = fix-y)giy)

We want to first show that the function F{x,y) is a Borel function in R2. For a set £ e R , let there be a set E e R2 defined by

E = {{x,y):x-yGE}

Since x — y is a continuous function of (JC, y), E must be open whenever E is. It is very easy to verify that the collection of all E ε IR for which É (as defined above) is a Borel set forms a σ-algebra on R. Again, if V is an open set in R and / is a Borel function on R, the set E = {x : f(x) E V} is a Borel set in R. Therefore

{(x,y): fix - y) £ V} = {(x,y): x - y e E) = E

is a Borel set in R2. Hence the function (x - y) —♦ f(x — y) is a Borel function. The function ix,y) —* giy) is also a Borel function in R2. Therefore the product fix ~ y)giy)iS a Borel function on R2. Now

dy I \F(x,y)\dx= \giy)\dy \f{x - y)\dx ■00 J —00 , / — 0 0 J —OO

= 11/11, llglli

We have used the translation invariance property of the Lebesgue measure to show that

/ \f(x-y)\dx = \\f\U J-tx

So F(x,y) £ LX(U2). Therefore in view of the Fubini's theorem ( / * g)(x) = f-o* f(* - y)8(y)dy exists for almost all x £ R , a n d ( / * g)(x) £ L'(R). This proves (i) and (ii) together. Now

/

oo «»00 r y*oo

\(f*g)(x)\dx< / / \F(x,y)\dy dx ■00 J —00 L» / —OO

= f UjF{x,y)\dx\dy = ||/||, \\g\U This proves (iii). D

Corollary 5. Let f,g £ L'(IR). Then

J(f * g)(w) = (J/)(w)( Jg)(w)

where F is the Fourier transformation operator.

Proof.

W*g)(w)= [ (J f{x-y)g{y)dy\eiwxdx

Then by Fubini's theorem we get

= J_m ( / / < * - yywxdx^j g(y)dy

Now

/

OO »00

f{x - y)eiwxdx = eiwy / f(x - y)eiw{x'y)dx ■00 J —00

= eiw> [ f{t)eiw'dt J -GO

By the translation invariance property of the Lebesgue measure,

JX/*g)(w) = f eiw> f nt)eiw'dtg(y)dy J~ OO J~ 00

/

OO /.OO

g(y)er>dy / / ( θ Λ ί oo y—oo

= ( Tf)M(Jg)(»>)

10 SOME BACKGROUND

An excellent proof of the following theorem is given by Hewitt and Stromberg [48, p. 397]. D

Theorem 7. For 1 < p < °°, let / E L\U) and g E LP(U). Then for almost all x E IR, f(x — y)g(y) and f(y)g(x — y) as functions of y, E L'(IR). For all such x define

(f*gX*)= [ f(x-y)g(y)dy

and

= [six-. JR

Then

and

(**/)(*)= / g(x~y)f(y)dy

(/ * g)W = (g * /)(*) a.e.

(/*e)U)ez/(R)

further | | / * g | | p < | | / | | , ||g||p.

Proof. Let q = -4γ, and let h E L«(IR). Then each of the functions f(x - y), g(y), h(x), are Borel measurable in IR2, and so also are their products taken two at a time and the function f(x — y)g(y)h(x). Now using Fubini's theorem, translation invariance of the Lebesgue measure and Holder's inequality we have

/ / \f(x-y)g(y)h(x)\dydx J — 00 J — 00

= / ΙΛΟ0Ι / \f(x-y)g(y)\dydx J—CO J— CO

= f \Kx)\ f \f(t)g(x-t)\dtdx J—CO J — 00

= I 1/(01 I \g(x-t)h(x)\dxdt J—CO J—CO

=s / l/WllliU-OllpllAWll,* J - 0 0

= 11*11, HAIL 11/11, < «

It follows that

/ \f(x-y)g(y)\dy J - 0 0

and

/ \f(t)g(x-t)\dt J -00

are both finite for almost all i £ R , and so ( / * g)(x) and (g * f)(x) are finite a.e.

I / ( / * g)(x)h(x)dy\ = \f (g* f)(x)Kx)dx \J— oo \J~oo

* yi,ii/n, HAH, Since the dual of L?(R) is i /(R), ( / * g)(x) G the dual of L«(R), (/ * g)(x) E L". Now in the duality notation we have

|<(/*sX*).AW>|fi ||*||„||/|| I \\n\\q

Ηί,Λ v ,n \<(f*g)(x).h(x))\ \\(f * g)(x)\\p = sup J ¡r—r L s y | p | | / | | i

w,=i ll«wlU

(/*,?)(*) £Ζ/(-°°,οο) and

\\f*g\\P*\\f\\x\\g\\P

The fact that (/ * g)(x) = (g * f)(x) a.e. is also proved by using the above duality results. This is because, by Fubini's theorem, we now have

<(/ * *)(*). AW> = ({g * /)(*), Kx)) VAG L«(-oo, oo)

Therefore

(/ * S)M = (i * f)(x) in £ ' ( - » , oo)

and so a.e. D

There are many interesting results on the convolution of classical functions. The interested reader may consult Hewitt and Stromberg [48] and Treves [101]. Treves [101] gives the proofs of Theorems 7 and 8 using Holder's inequality and the density property of the space TKW) into the space LP(R"), p > 1. This is a very interesting approach.

12 SOME BACKGROUND

1.3.1. Differentiation of the Fourier Transform

Theorem 8. Let / be a function defined on the real line such that /(<) and t /(f) are both absolutely integrable on the real line. Then

J¿(J / ) (HO = J[itf(t)}

The proof is simple. The theorem can be proved by using the classical result on the switch of the order of integration and differentation, as shown by Titchmarsh [100]. Theorems 6, 7, and 8 are also valid in IR" and the proofs of these theorems can be obtained in the same way.

I will develop some elementary properties of distributions and then will return to the theory of the Fourier transform of distributions. The discussion will throw some light on the work done by Schwartz and L. Ehrenpreis in this direction.

1.4. THEORY OF DISTRIBUTIONS

Before I turn to the Schwartz theory of distributions, I will consider some prerequisites to it. I will assume that the reader is familiar with elements of functional analysis.

Topological Space. Let X be a nonempty set. A collection τ of subsets of X is said to be a topology on X if the following properties (axioms) are satisfied by r.

1. φ G randX G τ. 2. If u\, M2 u„ belong to τ, then f]"=, M, G T. 3. Let {ua}a&A be an arbitrary collection of the members of τ, then |JaeA M« e τ ·

It is understood that the collection {ua}a^A in property 3 can be finite or infinite (countable or uncountable).

If T is a topology on X then members of τ are called open sets in X, and the pair (X, T) together forms a topological space. Often the topological space (X, τ) is simply denoted by X.

Example 2. Let X = {0,1,2,4}, and let τ be the collection of subsets of X given by

τ = {φ}, {0,1}, {1,2}, {1,0,2}, {0,1,2,4}

Prove that the collection τ does not form a topology on X:

{0,1}η{1,2} = { 1 } £ τ

So T does not form a topology on X. But the collection

τ = {ψ}, {0,1,2,4}, {1,2}

does form a topology on X. Verification of axioms 1, 2, and 3 is easy.

THEORY OF DISTRIBUTIONS 13

Now let X and Y be two topological spaces, and let / : X —» Y. The mapping / is said to be continuous if /~ '(V) is an open set in X for every open set V in Y. The mapping / is said to be continuous at a point x E X if the inverse image of every open set containing f(x) in Y is an open set in X.

A nonempty set X is said to be a metric space if there is a distance function p defined on it such that

1. 0< p(x,y)<°°Vx,y 6 1 2. p(x,y) = 0iff* = v. 3. p(x, v) = p(y, x) V x, y E X. 4. p(x,y) ^ p(x,z) + pU,y) (triangle inequality).

The set {y E X : p(x,y) < r} is defined to be an open ball with center at x and radius r > 0. Let τ be the collection of all sets £ E X that are arbitrary unions of open balls. Then τ is a topology on X. It is very easy to verify that τ satisfies the axioms 1, 2, and 3 for a topology. Therefore a metric space is a topological space.

1.4.1. Topological Vector Spaces

A topological vector space (or TVS) X is a vector space equipped with a topology T such that the operation of addition from X X X —» X and the scalar multiplication from C X X —» X are both continuous. Thus a topological vector space is a vector space whose topology is compatible with its linear structure. A normed linear space X is a topological vector space. If xa —> xo, ya —* yo, then

Ik, + ya - (*o + yo)\\ =£ Ik, - *0ΙΙ + \\y* - joll - 0

Therefore xa —> x0, ya —► yo => Ua + ?<*) —» *o + 3Ό- If ca —> c0 and xa —> JC0, then

\\caXa - C0X0\\ = \\(Ca - C0)xa\\ + IM*« ~ *θ)ΙΙ

^ \ca - col Ik, II + kol \Ua - Jfoll

Since xa —» JCO, it can be shown that there exists M > 0 such that ||*α || < Af, V jra E a neighborhood of xo. Therefore

\\caxa - c0xo\\ ^ k 0 - c0|M + kol Ika - JColl -» 0

as ca —> c0 and Jta —» Λ:Ο; then ca —> c0 and xa —> x0 => caxa —> coJCo-A subset A of a topological vector space X is said to be bounded if for any

neighborhood U of 0 E X there exists a constant £ > 0 such that kA C U. A neighborhood of a point is defined to be a set containing an open set that contains the point. Equivalently a subset V of a TVS X is said to be bounded if by suitable contraction it can be contained in any neighborhood of zero. More precisely for a given neighborhood U of zero there exists a λ > 0 such that

V<Z\U

14 SOME BACKGROUND

Definition. A subset A of a vector space V over C (field of complex numbers) is said to be convex if x,y G A and Va, ß G Ä+, such that a + ß = 1 implies that ax + ßy G A.

Example 3. {x : \x — XQ\ < r} on W is convex where

\x- Xo\ = ¿ U - χο,Ϋ l"/2

. 1=1

Example 4. {x : \\x\\ < b} in a Banach space is convex. Also the set

x : Σ&ί - xoif ¿=1

1/2

<¿>

in W is convex.

Example 5. Intersection of any family of convex set is a convex set. A subset A of a vector space V is said to be balanced if λΑ C A for all λ G C such that |λ| ^ 1. Taking λ = 0, we see that 0 belongs to any balanced set A. For example,

|JC : U| = y/x + x\ + x; l^p)

is a balanced set of W, but

{x : v/(*, - l)2 + (x2 - l)2 + · · · + (xn - 1)2 < p}

is not a balanced set of IR" if p < y/n. On V = IR the closed interval [1,2] is convex but not balanced. One can verify

that the intersection of a family of convex and balanced subsets in a vector space V is a convex and balanced subset of V.

A subset A of a vector space V is said to be absorbing if for every x E V, there exists a λ > 0 such that x G λΑ. Note that an absorbing set must contain the origin. The set [-2,0,5] is an absorbing set in U, but the set [1,2,5] is not an absorbing subset of U.

Any neighborhood of the origin in a topological vector space V is an absorbing subset of V. A family of open sets in a topological space (X, τ) is said to form a basis of the topology if every open set in X is the union of sets of the family. A family of neighborhoods of a point A: in a topological space is said to form a neighborhood basis if every neighborhood of x contains some set of the family. Two neighborhood bases at the point x are said to be equivalent if each neighborhood in one base contains a neighborhood in the other base. Let 7Ί and T2 be two topological bases in X. Then T\ and T2 are said to be equivalent if each open set in T\ contains an open set of T2 and each open set of T2 contains an open set of T\. For every x G X, let Bx be a

neighborhood basis at x, and let B = U*ex &χ· Then obviously 2 forms a basis for the topology on X. The topological space X satisfies first countability axiom if there is a countable neighborhood basis at every point of x E X. If there is a countable basis for X, then X is said to satisfy the second countability axiom.

Consider a topological vector space X, and define a translation mapping x —> X+XQ. Clearly it is a mapping from X onto itself and is continuous, and its inverse mapping x —» x - x0 is also a continuous mapping from X onto itself. Therefore the translation mapping is a homeomorphism from X onto itself. It maps an open set into an open set. Once we know the neighborhood basis at the origin, we know the neighborhood basis at any point JCO Ξ X by simply making use of the translation mapping. Thus the topology of a topological vector space can be completely determined by knowing a neighborhood basis at the origin. If T is a linear functional on a topological vector space X, then the continuity of T at any arbitrary point x <=> continuity of T at 0. The proof is very simple.

A subset Y of X is defined to be a topological subset of X if y is equipped with the topology induced by X on Y, which really means that a set M C Y is said to be open set in Y iff there exists an open set B C X such that

M = Y C\B

Let there be two topologies T\ and τ2 on a set Y. We say that the topology T\ is stronger than the topology τ2 if every element (open set) of τ2 is contained in Ti. We say that two topologies T\ and τ2 on X are equivalent if τ\ is stronger than τ2 and also τ2 is stronger than τ\.

Definition. A topological vector space X is said to be locally convex and called a locally convex space if there exists a neighborhood basis at 0 in X whose every element is a convex set.

A functional y on a vector space X is said to be a seminorm if it satisfies the following conditions:

i. γ(φ) > 0 V φ E X. ii. γ(αφ) = |ο|γ(φ) V φ E X and a E C.

iii. y(<p + ψ) < y(<p) + γ(ψ).

In fact the condition i is redundant. If we put a = 0 in condition ii, we get γ(0) = 0. Likewise, by putting φ = φ - ψ in condition iii and interchanging the roles of φ and ψ, we get y( x = 0 becomes a norm.

Theorem 9. Let y be a seminorm on a linear space X. Then the γ-ball By(r) = { i £ X : y(x) < r} is (i) convex, (ii) balanced, and (iii) absorbing.

16 SOME BACKGROUND

Proof, (i) Convex: If x, y G By(r), then for a G [0,1],

y(ax + (1 - a)y) < y(ajr) + γ((1 - a)y)

< | α | γ ( χ ) + | 1 - α | γ ( ? )

< a-y(x) + (1 - α)γ(>)

< ar + (1 — a)r

< r

(ii) balanced: Let a E C such that |a | < 1 and JC G By(r). Then

■y(ajf) = |α |γ(*) < r

\JaBy(r)CBy(r) if |o| s 1 a

(iii) absorbing: λθ E ß r ( r) . If y(x) = 0 for x Φ 0, then JC G ß r(r) . Therefore for a nonzero JC G V such that γ(χ) Φ 0, choose A* = - ^ . Now γ(λ^) < r, that is, \xx G Äy(r). D

Let £ be an absorbing set in a vector space V, and let x be any point in V. Define a functional μ£ by μΕ(χ) = inf{A > 0 : A ~ ' J C G £ } = inf{A > 0 : x G λ£}. Clearly μ£(0) = 0 in that, if A, is a sequence of positive number tending to zero, then inf, A, = 0. Therefore

0 < μ£(0) < inf{A, > 0 : 0 G A,£}

0 < μ£(0) < 0.

Hence μ£(0) = 0. The functional μΕ : X —> [0, °o) as defined above is called the Minkowski functional of £ f 1 ].

Theorem 10. Let £ be a convex balanced and absorbing subset of a vector space V. Then the Minkowski functional μΕ(χ) of the set £ is a seminorm over the vector space V.

Proof. For any x,y GV, let

μ(χ) = inf{A > 0 : x G λ£}

μ(>>) = ίηί{μ > 0 : y G μΕ}

We can choose nonnegative numbers A and μ satisfying

με(χ) ^ A S μΕ(χ) + e and x G A£

MEOO ^ μ S μ ^ ) + e and y G μ£

By virtue of convexity of E,

A + μ A + μ A + μ

Therefore

με(χ + y) ^ λ + μ < μΕ(χ) + e + μΕ(γ) + €

Since c is an arbitrary positive number, we have

με(* + y) ^ με(*) + MEOO-

Again,

μΕ(χ) < λ < μ£(Λ) + €, * G λ£

If c is a complex number (excluding the trivial cases c = 0) and η is an arbitrary positive number < μΕ(χ) Φ 0, then by definition

x £ (μΕ(χ) - η)Ε

Therefore

ex $ (\ε\μΕ{χ) - \ε\η)Ε

If ex G λΕ for λ > 0, then λ > (ΜμΕ(χ) — \c\r¡). Therefore με^χ) = inf λ s I C ^ E U ) — |C|T). Since η is arbitrary, we get

μΕ(ο:) > μΕ(χ)Μ (1.15)

Also

ex e A|c|£ V λ satisfying λ > μΕ(χ)

μΕ^χ) < inf A|c| = |c| inf λ = ΜμΕ(χ) (1.16)

Combining the two inequalities (1.15) and (1.16), we have

μΕ(οχ) = ΜμΕ(χ)

The result follows from (1.16) if μΕ(χ) = 0. D

Definition. The family P — {pa}aeA of seminorms defined over a topological vector space X is said to be separating if for each nonzero x G X there is at least one seminorm Ρβ e P such that ρβ(χ) Φ 0.

A TVS X is said to be Hausdorff if every two different points in X have disjoint neighborhoods.

18 SOME BACKGROUND

Theorem 11. The topology of a locally convex topological vector space X is equiv-alent to the topology generated by a metric if and only if X is Hausdorff and has a countable local base.

Proof, (i) Assume that X is metrizable, namely that there exists a metric p(x, y) that generates the topology of X. Then the set of balls {Rn}°H=i where Bn = {x e X : p(0,x) < ¿} are convex, balanced, and absorbing and form a countable base at 0 for a Hausdorff topology, (ii) If X is Hausdorff and has a countable local base {£n}™=i at 0 such that each of B„ is convex, balanced, and absorbing, define the seminorm p¡ on the space X to be the Minkowski functional of B¡. Set

, , v> i pa* -y) w ._ v

Now

if 0 < a < b \+a \+b

P¡(x - y) _ Pilx -z + z-y] 1 + pi(x - y) 1 + pi[x -z + z-y]

p¡(x - z) + p¡(z - y) 1 + pi(x - z)+ p¡(z - y)

Pi(x - z) p,(z - y) 1 + pi(x - z) 1 + pi(z - y)

and

p(x,y) < p(A:,z) + p(z,y)

Therefore p is subadditive and p(x,y) = 0 => x = y as {ρ,}°1] is separating. Next to show that p(x,y) = p(y,x) ^ 0 is trivial. The fact that the topology

generated by p(x,y) is compatible with the local structure of X follows from the properties of seminorms p¡. Now let

U„ = ixGX:d(0,x) 2 " /

p : X X X —* R is continuous as the series representing p(x, y) converges uniformly. Since each of the terms ^ '^"^L is continuous, the converges must be continuous function of x, y. So Since each of the terms ^",^, is continuous, the metric p(x, y) to which this series

U„ = LeX:piO,x)<±\

is open in (X,B). Clearly U„+\ C B„ because, if x $ Bn, then

p(0, x) s -— y (a contradiction)

Therefore {U„} forms a local base for the topology of (X, B). O

Theorem 12. Let V be a topological vector space. A subset B of V is bounded if and only if every sequence contained in B is bounded in V.

Proof. If B is a bounded set in V, then every sequence in B is bounded (obvious). Now assume that every sequence of B is bounded in V. We wish to show that B is

bounded. Assume that B is not bounded. Then corresponding to any positive integer n and a neighborhood U of zero, we can find an element x„ £ B such that x„ £ «Í/. Therefore we are able to find a sequence {xn}™=\ in B that is not bounded, and this is a contradiction. D

There are situations where the topology of a topological vector spaces can be generated by a norm. The topology generated by a norm will be equivalent to the topology of the topological vector space under consideration. Such topological vector spaces are said to be normable.

Any ball in a normed linear space is bounded. So in a normed linear space there exist balls which are neighborhoods of the zero element that are at the same time bounded. This property is the characteristic of normable spaces at least among Hausdorff locally convex spaces.

Some topological vector spaces are not metrizable. An example of this type of a TVS is the Schwartz testing function space D equipped with the usual topology, which I will discuss in the next section. An excellent proof of this fact is given by Shilov [88]. Since D is not metrizable, it is not normable either. Theorem 13 below gives a characterization of locally convex spaces that are normable. We will need the following lemmas to prove it:

Lemma 1. (i) Let U be a convex subset of a vector space V. Then the set U0 defined by V0 = U|A|se λί/ is also a convex subset of the vector space V. (ii) t/0 is balanced.

Proof. If x and y are two elements of the set U, then for 0 < a < l ,ax + (l -a)y G U. Therefore Kax + λ(1 — a)v ε λί/. Let K\x and kiy be two representative elements oiX.\U and A2i/, respectively, forO < |λι|, |λ2| both < e. Our claim is that the element

a\\X + (1 — a)\2y

belongs to the set

(αλ, + (1 - a)K2)U

20 SOME BACKGROUND

where 0 < a < 1. If we put

A[ßx + (1 - ß)y] = αλ,χ + (1 - a)k2y

then

A(l -β) = ( Ϊ - α)λ 2 } =* A = «A, + (1 - α)λ2

Further note that if |λ] |, |λ2| ^ e, then

|αλ| + (1 - α)λ2 | i£ €

Therefore i/0 is a convex set. (ii) Let k £ C such that \k\ < 1. if λ G C such that |λ| < e. Then *λ e C,

satisfying \k\\ ^ e.

kU0 = IJ *λί/ |A|£«

= U μί/CÍ/o luis«

Therefore i/o is balanced. D

Corollary 6. Let X be a locally convex Hausdorff TVS. Then there exists a subset i/o of X that is convex, balanced, and absorbing.

Proof. Since X is a locally convex TVS, every neighborhood of 0 contains a convex neighborhood of 0. Again the mapping

ax:CxX ->X

is continuous. Let V be a neighborhood of zero. There exists e > 0 and a neighborhood U of 0 such that

ax £ W a satisfying \a\ ^ e and x & U

that is, all G V. The set U is chosen to be a convex set by choice as X is locally convex. Now let

i/o = (J ail l«|S€

Each αί/ is a neighborhood of 0, so i/o is a neighborhood of 0 as well. Therefore i/o is absorbing, as a neighborhood of zero in a TVS is absorbing. By Lemma 1, i/o is convex, balanced, and absorbing. D

Theorem 13. Let X be a locally convex and Hausdorff TVS. The topology of X can be generated by a norm if and only if its zero vector has a bounded neighborhood.

Proof. Let X be normable with a norm. Then the unit ball {x : \\x\\ < 1} is a bounded neighborhood of 0. If U is a bounded neighborhood of 0 in a TVS X that is Hausdorff and locally convex, we can find an open neighborhood i/o of zero contained in U such that i/o is convex balanced, absorbing, and bounded. Let p be the Minkowski functional defined on i/0. By the mode of construction of the Minkowski functional, if p(x) = 0, then x £ λί/ο for any λ > 0. Now, for any neighborhood of 0, we can choose λ > 0 appropriately so that λί/ο is contained in the neighborhood of zero. So x belong to every neighborhood of zero. Since X is Hausdorff two points 0, and x cannot be contained in every neighborhood of zero unless x = 0. Therefore p(x) = 0 =Φ· x = 0. So p is a norm. D

For further details about the normable and metrizable TVS one can look into Treves [103].

Definition. The support of a complex-valued function / defined over a set Ω is the closure of the set { j t E f l : φ(χ) Φ 0} in the topological space Ω. That is, it is the smallest closed set containing the set {x G Ω : φ(χ) Φ 0}. Define a function f{x) over the interval [0,1] by

,, . _ J 1, where x is rational ri 171 where x is irrational

Here the support of / = supp / = the closed interval [0,1 ]. If, however, we restrict to the case that f(x) is defined over the interval (-1,1) by (1.17), then supp / = (—1,1) and not [-1,1]. Now consider the infinitely differentiable function φ(χ) such that

φ ( * ) = / ε χ Ρ ϊ 4 ρ when W< 1 10 when \x\ > 1

Then the supp φ is the unit ball { j t £ R " : \x\ < 1}.

1.4.2. Locally Convex Spaces

Let Ω be an open subset of R". We denote the set of functions defined on Ω and continuously differentiable up to order m by C(m)(Ci). Now assume that AT is a compact subset of Ω. Then the set of functions that are continuously differentiable up to order m and have support contained in the set K will be denoted by CJ¡f\Sl). It is easy to see that C(m)(n) and c£°(ft) both are vector spaces and that Cp\Sl) C &η\Ω).

The space of infinitely differentiable complex-valued functions defined over Ω will be denoted by C°(Cl) = {\,^0 C(m\Q,). The set (Γ£(Ω) stands for the set of infinitely differentiable complex-valued functions defined over Ω and having support contained in K. The space CJ(ft) will stand for the set of complex-valued and

22 SOME BACKGROUND

infinitely differentiable functions defined over Ω having compact supports in Ω. Recall that a topological space is a locally convex space if its topology has a local base whose members are convex sets. Each of the foregoing vector spaces can be equipped with a topology that will make them locally convex TVS. As an example consider the space C^\D,). Define the set of seminorms {·ν*}™=0 by

γ*(φ) = sup \όαφ\ (1.19) xGK Ms*

The collection of seminorms {γ*}™=0 IS a separating collection of seminorms. The

space C%"\ñ) becomes a locally convex Hausdorff TVS when it is equipped with the topology generated by the sequence of seminorms {γ*}"=0 defined by (1.19) as follows: We define the open neighborhood basis about the origin as the collection of balloons centered at the origin by

{ψ· *M<P) < et, * = l ,2 , . . . , r}

where et, €2 er are arbitrary set of positive numbers and v\, v¡.,..., vr are integers lying between 0 to \m\, and likewise r is an integer lying between 0 and m. Denote this class of balloons by ßo. Since the space C(™\Cl) is a vector space, the open neighborhood basis about a point ψ will be given by B0 + ψ, which we denote by Βφ. Then UXECW) ^χ forms a basis for the topology of the space C"(Sl) with which we equip it. One can verify that the space C%(il) is a locally convex, Hausdorff, and "complete (sequentially complete)" TVS. Convergence to zero of a sequence {<p„}"=, in this topology is equivalent to the uniform convergence of each of the derivative sequence {φ®}™^, to zero for each |i'| = 0,1,2, . . . , m over K and so uniformly over Ω. This space is also metrizable because its topology can be generated by the metric

ρ<^) = Σ ? 1 + γι.(<ρ-ψ) ( , · 2 0 )

|ι|=0

For details of the basis of a topology, the reader is referred to Bremmerman [8], Friedman [42], and Kelley [55], and Zemanian [110]. The set of continuous linear functionals over the space C^iil) is denoted by (C£(il))'. Obviously it forms a vector space. If f,g €Ξ (ί7£(Ω))' and a, β are complex constants, we define the functional 0 / + ßg by

(af, φ) = a(f, φ)

</ + S, = ( / » + <£.<?> V * e C Z ( f t )

An example of a function belonging to the dual space C^iíl) is the functional 8(x — a) where

(δ(χ - a), φ) = φ(α) V φ & (7ΐ"°(Ω)

1.4.3. Schwartz Testing Function Space: Its Topology and Distributions

Let Ω be an open subset of W and K\, K2, K3,.. . be compact subsets of Ω such that

Kx C K2 C · · · C Kn ...

and

Such a construction is always possible [Yosida, 108]. We have already discussed the space C^(fl) and the space CQ(ÍI). When the space

CQ (Ω), which consists of infinitely differentiable complex-valued functions defined over a subset Ω of W having compact supports in Ω, is equipped with the topology as described by Laurent Schwartz [87], the space is named as ΤΚΩ). Clearly

VKl({l)COKl(Cl)COK,(il)...

and

CO

Turn = u ©*,.(«) 1=1

Bremermann [8] describes briefly the topology of the space D(R") described by Laurent Schwartz [87] as follows:

Let €j be a monotone sequence of positive numbers tending to zero, and let {my} be a sequence of positive integers monotonically increasing to 00. The neighborhood basis B of 0 is defined as follows: B consists of the sets

V(W,{e;}) (1.21)

= {φ : φ G TKW), 1<?α«ρ(ί)1 < e,; V \a\ < m¡, t g Ω;}

where Ωο = 0 and Ω; = {t: \\t\\ < j}. B forms a neighborhood basis at the function φ = 0 and thus generates a topology for 2XR"). If {<p„}™=1 is a sequence in ΊΧβ.") tending to zero as v —► 00 in the topology of TXU") as described above, then there exists a compact subset K of W such that (1) the support of each of the elements <pv(t) is contained in K, and (2) each of the sequences {<pt*'(')}™= 1 tends to zero uniformly over K and so over W as v —> 00, for each k = 0,1,2 The uniformity here is assumed with respect to t and not with respect to k. The proof of this fact is easily obtained by the method of contradiction. The convergence in D(R") is very strong. There exist sequences of functions in ΊΧβ.") converging to zero along with all their derivatives uniformly on IR", yet the sequence may not converge to zero in T>(W) as the following example shows.

24 SOME BACKGROUND

Let φ be the function as defined by (1.18). Then φ €Ξ ΊΧΜ"). Define a sequence {<?,(')}:=, by

φΛΟ = -<ρ(ί)· v

Then φ„(ί) -» 0 in ©(IR") as v -> °°. But the sequence

<M0= -φ(-)

1 /Ί h t„\ = -φ[-.- -

v \ v v v I does not converge in D(IRn) as v —► °°. This is because there may not exist a compact set K E U" that contains the supports of each of φν(ι). There is, however, an equivalent way of describing the topology of the space D (as described by Schwartz [87], Zemanian [110], and Yosida [108]). We will now explain it.

Let Ki, Κ2, ΛΓ3 be the sequence of compact subsets of Ω e IR" such that

KXCK2CK3...

and CO

Recall that

DJfl(n)cDjfj(n)cDjr,(n)... and

00

TKÍI) = U £>*, (Ω) i = l

The topology on each of the spaces Ί)χ(Ω) is described by the sequence of seminorms {">Ί(<ρ)}Γ=ο a s defined by (1.19) and the space Τ>κ(Ω) is metrizable by the metric (1.20). It can be easily seen that the topology of T>K¡ (Ω) is the same as that induced on Όκ, (Ω) by Όκi+1 (Ω) for each i = 1,2,3,4

A natural way to topologize the space ϋ ( Ω ) is that if V is an open set in TXÜ.), then V Π Όκ(Ω) is an open set in 2)^(Ω) for any compact set K E Ω. Therefore, if V is an open set containing the origin in the space ΊΧίϊ), then V Π 23^,(Ω) is defined to be an open subset of Τ>κ,(Ρ) containing the origin for each / = 1,2,3 When topologized in this way 23(Ω) is called as the inductive limit of O/c¡(Ciys (see Yosida [108, p. 28], Shaefer [85, pp. 56-57]). When ΊΧΩ) is topologized in this way, the embedding of 2?*(Ω) in TXVL) is continuous, and this is a very natural thing to expect.

Theorem 14. Let D(fl) be the inductive limit of 2)^.(Ω). That is, K,'s are compact subsets of Ω G W and

00

Ω = {JK¡

K\ C Kj C K^ C Ω. Then the convergence of a sequence {φ^} to 0 in 2)(Ω), namely lim*—«, <p* = 0 in 2)(Ω) means that

i. There exists a compact subset K of Ω such that supp ftC^Vt= 1,2,3, ii. For any differential operator V, the sequence {2y/i(jc)}J°=1 converges to 0

uniformly on AT.

Proof (by Yosida [108]). We need prove only (i), since (ii) is merely a trivial conse-quence of (i). Assume that (i) is not true. Therefore there exists a sequence {J:(*')}"= , of points of Ω having no limit points in Ω and a subsequence {φ* (χ)} of {φ*(*)} such that <pic (XJ) Φ 0. Then the seminorm

oo

ρ(φ) = Σ 2 sup

Xj G Kj — Kj-\

Ko = Φ

defines a neighborhood U = {φ G ^ ( Ω ) ; ρ(φ) < 1} of 0 of ΊΧ$Ϊ). This is because

U Π VKl(fl)

c{*:<pG^,:|MI-^} and

U Π ΌΚι(ίϊ)

t-í e n u n « 0<M*i)U<M*2)l)l C ^ φ : φ G D^2 : IMI < max - '- \

and so on, where IMI = sup^eR |φ(χ)|. Therefore U as described above is a neighborhood of 0 in 2?(Ω). But none of the

<PiCj belong to U as p(<Pk(x)) ^ 2, so <p*; is not eventually in U. Therefore φ*; does not tend to zero, a contradiction. D

<p(x) <PkÁXj)

I will now show that the space T> = ΤΧ.Ω) is not metrizable. The following counterexample is a special case of the counterexample given by Shilov [88]. Define

26 SOME BACKGROUND

a double sequence function {φ„,μ}™μ=1 in 25 as

f i g - ^ i |,| < „ <Ρκμ(0 = < "

I 0, |i| > v

Clearly φν,μ(0 E 25, and the support of φν,μ(ΐ) is contained in the interval [— v, v], Assume that p is the metric describing the topology of the space 25. Now consider the following sequence of sequences

<Pi.i.<Pi.2.<Pi.3··· ->0 in 25

<P2,1.«P2,2.<P2,3··· - » 0 in 25

<P3,1.<P3.2.<P3.3··· - » 0 in 25

Choose an element φι*, from the first horizontal sequence so that p(0, <Pijt,) < e. Make sure that φ^, lies to the right of <piri in the first horizontal sequence. Choose an element <p2,¿2 from the second horizontal sequence so that p(0, <P2,*2) < f. taking care that ψι^ lies to the right of the element ^2,2· Thus we are able to choose a sequence Wv.kX=\ suchthat

P ( 0 , as v —» 00. But this is a contradiction, for the support of <pv¿y is [ — v, v\. So as v —* °°, the support of the sequence <?„*„ gets unbounded. Therefore the sequence {<p„,*„}"=1 does not converge in T>. This contradicts our assumption that the space T> is metrizable.

A distribution is a continuous linear functional on D(M"). When n = 1, we denote the testing function space by T>. Sometimes a distribution is also called a generalized function. I now present some examples of distributions.

Definition. A regular distribution is a distribution generated by a locally integrable function / , and this distribution is also denoted by / ,

L </,<*>>= / f(x)<p(x)dx

Put differently, the distribution / is the functional when operating upon the testing function φ generates the same number as the integral

f /ΟΟφΟΟΛτ - f fW<p(x)dx J—00 Ja

where the support of <p(x) is the interval [a,b]. The support of φ varies with φ. Linearity of this functional is trivial. To verify continuity, let <pv —» 0 in D as v —> °°,

and let the supports of all <p„ be contained in an interval [A,B]. Then

K/.<P,>I= / f(x)<PÁx)dx\^e \f(x)\dx \JA I JA

\φν(χ)\ S e as v —► oo

Since e is arbitrary, it follows that

if, φν) —► 0 as v —* o>. The example of a regular distribution over 1>(U") can be given similarly. A distri-bution that cannot be generated by a locally integrable function is called a singular distribution. The Dirac δ function is a very simple example of a singular distribution,

<δ(ί-α),φ(ί)) = φ(α)

Linearity and continuity of this distribution are trivial, pv £ is another singular dis-tribution, and it is defined as follows:

pv l-, <p(x)\ =(P)[ *Qdx. (1.22)

The fact that the right-hand side in (1.22) exists is very easy to prove. Let the support of φ be contained in the interval [—a, a]. Then

X I J-a X

= (P) r iM-m^

= {P)r«*-«®dx+{nr J-a X J-a

J-a

Now define a C°° function ψ(χ) by

J sii

Ιφ'ι ψ(χ) = { , * ' 1.23

(0), if x = 0

Therefore

Now

ρυ-,φ(χ)\ = f φ(χ)άχ (1.24) ΛΓ

*w = - Γφ'(ί)Λ x Jo

28

Thus we obtain

|i/í(jc)|<sup|<p'(í)l

SOME BACKGROUND

(1.25)

The linearity of pv ¿ follows from (1.24), and the continuity from (1.25). Therefore pv\ £ £>'.

Pseudofunction l+(i)/"3^2 where

1 + ( ' ) = {o , ,<0 The generalized function pseudofunction 1 + (t)t'3^2 is also denoted by Pfl + (t)t~3^2. The function l + (r)i-3^2 does not define a regular distribution because the integral Jo t~3/2(P(Odt, ψ G T> is divergent in general. We therefore generate a distribution from 1 + (f)i~3/2 by using Hadamard's technique of splitting a divergent integral into convergent and divergent parts. The convergent part of the integral /0°° ί _ 3 / 2φ(ί)Λ denotes the expression

</>/ι+(ί)/-3/2,φ(/)> ν φ ε ζ > (1.26)

If b is large enough so that φ(ί) = 0 for / > b, then the expression in (1.26) becomes rb

lim Fp / Γ3/2φ(ί)ώ

rb = lim Fp / ( " 3 / 2 [ # ) + /ψ(ί)]Λ from (1.23)

= lim Fp e-.o+

2φ(0) _ 2<p(0) h

Jt Jt

We throw away the divergent part and define

2φ(0) Jo Jo \Jt \/b

Therefore 2<p(0)

Jo V ' v&

Now the linearity of this functional is obvious. For {φν} —► 0 in £>, assume that the support of <p„(f) is contained in the interval [—b, b]. Therefore

Jo Jt Jb

Jo Jt \ψ'ν(χ)\ 2jl>->0 a sv ->°°

This proves the continuity of Pf\ + (t)t~i/2. Similarly we can define many other pseudofunctions by using the technique of Hadamard. Some of the well-known pseudofunctions are Pfl + (t)t~i2n+X)/2, Pfl + (t)lnt, Pf1^, and so on. The details can be seen in Zemanian [ 109]. A distribution / is said to be zero over an open set Ω if (f, φ) = 0 for all φ G TKM") whose support is contained in Ω. Two distributions / and g are said to be equal over an open set Ω if

(f,<p) = {g,f>) V<pEüdRn)

with support in Ω. The support of a distribution / is the smallest closed set outside which the distribution vanishes. The union of all open sets over which a distribution vanishes is called the null set of the distribution. We can see that the support of a distribution is the complement of its null set. The support of 6(f) is the set {0}, and the support of the regular distribution 1 + (f - 1) 1 + (2 - /) is the closed interval [1,2].

Theorem 15. If a distribution is equal to zero on every set of a collection of open sets, then it is equal to zero on the union of these sets.

Proof of this theorem is rather complex. Readers interested in the proof may look into Zemainian [109].

1.4.4. The Calculus of Distribution

The power of distributional analysis rests in large part on the facts that every dis-tribution possesses derivatives of all orders and that differentiation is a continuous operator in this theory. Distributional differentiation commutes with many operations such as limiting operations, infinite summation, and integration.

Definition. We say that a sequence {/„} of distributions converges to a distribution / weakly or in the weak topology of 2?'(Ω) if the sequence (/*, φ) converges in the topology of C to (/, ψ) as k —» °°. Since (sin nx, φ(χ)) = JT«, sin ηχφ(χ) dx —* 0 as n —»oo. Therefore the regular distribution sin nx —> 0 weakly as n —► °°. The regular distribution generated by ^f- —► δ(χ) as n —► °°. This is an example of a case where a sequence of regular distributions converges to a distribution but the corresponding limit of ^215* in the sense of function as n —* °° does not exist.

irx

Consider the sequence

f o, id ¿ i v2, u\ < i

lim(/„, ψ) fails to exist whenever φ(0) Φ 0. This sequence is an example of an absolutely integrable function that tends to another absolutely integrable function but the distributional limit does not exist. Now consider another example of a directed

30 SOME BACKGROUND

sequence

M) \0. k l> i v fl/v

<fv.9)=~ <P« dt - φ(0) = <δ(ί), φ) 2 J-\/v

as v —> oo. That is, /„ —► δ(/) in T)1 but the function sequence /„(/) —♦ 0 a.e. as v —♦ oo. In this example both limits exist, but they do not correspond.

Theorem 16. The space Χ>'(Ω) is weakly complete. Let {/„}^=| be a sequence of distributions in 2?'(Ω) such that (/,,(/), φ(/)), φ e ΊΧΩ) is a Cauchy sequence in C. Then there exists a distribution / such that lim„_oo(/i,(/), <p(r)) = (/, <p) V (Ω).

Proo/. Since </„,φ> is a Cauchy sequence in C it must converge. Let us define a functional / by

\ίιηβν,φ) = (/,φ) (1.27)

This functional / defined by (1.27) is linear, and we now proceed to prove its continuity on ΊΧίϊ).

Let K be a compact subset of Ω, and let / be an open set containing K. The space

ΌΚ(Ι) = 2?ΛΤ(Ω) C Ζ>(Ω)

The topology of the space 2?κ(Ω) is generated by the separating collection of semi-norms

γ„(φ) = sup \da<p(t)\ |a|Sm

(6/

The topology of Τ>κ (Ω) can likewise be generated by the norm

*- ; 2m 1 + γ„(φ)

which we denote by ||φ||. The space Τ>κ(1) is really a Banach space, and each of the sequences {fv, φ) is

bounded. Therefore, by the principle of uniform boundedness, ||/„|| is bounded. In other words, there exists a constant c > 0 satisfying ||/ν| | ^ c. Now, using the property of a NLS, we have

|< /„ .|^IIMIIMI

\{/ν,ψ)\^θ\\φ\\

PRIMITIVE OF DISTRIBUTIONS 31

The sequence /„ is bounded over ©*(Ω)

(/,<p) = \im(fv,<p)

and therefore

|(/»|^|MI So / is bounded on ϊ?κ(Ω); / is continuous on 2)κ(Ω). Since K is arbitrary compact subset of Ω, it follows from the definition of the inductive limit topology of TXfl) that / is continuous on ΊΧίϊ). Thus 2?'(Ω) is weakly complete. D

Definition. We say that a sequence of distributions {/„}%, tends to / in the strong topology of £>'(Ω) if for every bounded set B of Σ>(Ω), {/„, ψ) —► (f, φ) uniformly V φ G B where B is an arbitrary bounded set of ΤΗ,Ω,). One can immediately observe that convergence of a sequence of distributions in the strong topology of 2)'(Ω) implies its convergence in the weak topology of 2?'(Ω).

1.4.5. Distributional Differentiation

The distributional derivative d"f of a distribution / is defined as a distribution that assigns the same number to a φ £ T> as / assigns to ( - 1)|α|<9α<ρ. It is given by the formula

<<?"/» = </,(-l)'<V V<p e ϋ(Ω). (1.28)

Recall that the expression θαφ stands for partial derivative of φ with respect to X\,X2 xn and of order \a\ = at\ + ctj + ■ ■ · + a„. The definition (1.28) is coined in analogy to the integration by parts in the classical analysis. Let

u,\ J1· ' - ° A ( 0 = \ 0 , t<0

Then h(t) is locally integrable and the distributional derivative of h(t) is 6(r). For

<DA(/),<P) = <MO, -<P ( 1 ) ( 0 )

= - ¡ φ'(ί)ώ = φ(0) = (δ,φ) Jo

we have

Dh(0 = 8(f)

1.5. PRIMITIVE OF DISTRIBUTIONS

The primitive of a distribution / is another distribution / ( _ I ) such that D / ( _ 1 ) = / . The primitive of δ(ί) = h(t) + c. There are infinitely many primitives of a distribution,

32 SOME BACKGROUND

and any two of them differ by a constant. We now state a theorem without proof and interested readers may look into the proof given by Zemanian [109].

Theorem 17. Every distribution on IR has infinitely many primitives defined by

where <po E T> such that / Γ . <Po(t)dt = 1 and K = / " „ <p(t)dt and χ = φ - Κψς,. Each primitive is also a distribution. The difference between any two primitives of a distribution is a constant given by

C = </f\«,> -if}-». 9o)

See Zemanian [109, pp. 72-78].

Theorem 18. Locally, every distribution is a finite order derivative of a continuous function. More precisely, let / be a fixed finite closed interval in IR', and let / be a distribution defined over a neighborhood of / . There exists a nonnegative integer r and a continuous function h(t) such that

</, φ) = (hr+2, φ) V φ £ TKJ) [Zemanian 109, p. 92]

Corollary. Let / be a distribution over IR with a bounded support. There exists a nonnegative integer v and a continuous function h(t) such that / ( / ) = A(y+2)(f) for all / [Zemanian p. 93].

1.6. CHARACTERIZATION OF DISTRIBUTIONS OF COMPACT SUPPORTS

We define the testing function space E{R") as the collection of infinitely differentiable complex-valued functions defined on W equipped with the topology generated by the open neighborhood basis

V(m,K) = {φ\ψ& E(R"),ym.K{<p) < e)

where ni's are nonnegative integers, e's are arbitrary positive numbers and K's are compact subsets of IR"

7mx(<*>) = sup |<9a<p| \a\Sm

A sequence {φ„}"= χ converges to zero as v —* °° if and only if each of the sequences {<PÍ*'(')} tends to zero uniformly on every compact subset of W. Distributions of compact support belong to E'(Rn), and conversely, any element of E'(W) can be identified as a distribution of compact support. This is because the space D((R'1) is a subspace of the space E(R") and because THU") is dense in E(W). The identity

CONVOLUTION OF DISTRIBUTIONS 33

map i : TKU") -» E(W) is continuous. Therefore the restriction of / £ E'(W) to IKW) is in I>'(W). Any element of E'(U") can be identified by an element of T>'(W) uniquely and hence E'(U") C T>'(W). We now give below a theorem that gives a characterization of E'(W).

Theorem 19. Every distribution / £ E'(W) has compact support.

Proof. Let K be the support of / . If K is not compact, then its support must be unbounded. Hence there exists a sequence of points ίμ £ K that has no finite accumulation point. With each ίμ as center, we draw a ball Βμ that does not intersect any of the balls Bv for μ < v. Thus we are able to draw balls with centers at ίμ such that the balls are all disjoint. Since each of the points ίμ belong to the support of / , we must be able to find φμ with support in ball Βμ such that (/, φμ) Φ 0. Now we choose constants εμ such that

<V</, φμ) = 1

that is, (/,εμφμ) - 1. Set

00

φ = Σεμφμ (1.29) μ=1

Since any compact set K intersects only finitely many balls the series in (1.29) is convergent and represents a function in E{W). Now

limN ( f, 53cß<Pß ) = (f- 

lim/v-,οο N = {f, φ), a contradiction. D

Therefore the support of any arbitrary / £ E'(W) must be compact.

1.7. CONVOLUTION OF DISTRIBUTIONS

In this section I will assign a meaning to the convolution of two distributions in a way analogous to the convolution of two functions, and the result will be used in the subsequent section on the Fourier transform.

Theorem 20. Let / £ T>'(W) and λ £ E(R") such that λ = 1 in an open set containing the support of / and zero outside a larger set. Then

< / » = </.λ<Ρ>

In other words, the value of (f, ψ) depends upon the value of ψ in a neighborhood of the support of / .

34 SOME BACKGROUND

Proof. Let λ be an element of ΤΧ$Ιη) as stated in the problem. Therefore, for all φ ε ZKR") and / e O'(W), we have

</.<Ρ> = ( / . ( 1 - λ ) φ + λφ>

= </,(1-λ)<ρ> + (/,λφ)

(1 — λ)φ is zero in an open set containing the support of / ; namely the support of (1 — λ)φ is contained in the null set of/. Therefore {f, (1 — λ)φ) = 0. Hence

</,φ> = </,λφ> D

1.8. THE DIRECT PRODUCT OF DISTRIBUTIONS

The operation of the tensor product or the direct product of distributions arises in the development of the convolution of distributions. This section presents some very essential features of the direct product to be used in the development of the theory of convolutions.

Let <p(t, T) £ D(R2) and /( /) G £>,' and g(r) G V'T.

Definition. Let / , g G T>'. Their direct product f X gis defined to be distribution in T>'tT or in V(U2) by

{f X g, <p(t, τ)> = (f(t), (g(T), <p(t, T))> V <p(t, T) G IXU2)

Justification of definition (1.30) can be given as follows: Let

<Kt) = (g(T)Mt,T))

Show that ψ(Γ) £ T>,. It is evident that ψ(ί) has compact support. To show that

Ψ'«) g(T),-<p(t,T) at

we have to show that

<ρ(ί + Δ/, τ) - <p(t, τ) θφ(ί, τ) Δί *

in DT as Δ/ —> 0. To prove this, we have to show that

<p(m)(/ + Af,T)-<p(m)(f,T) Δ/

- < p ( m + 1 ) ( i , T )

= _L /"+Δ' f ^ V o At J, [ dxm+

<y"+l<p(^T) <9(,π+1)<ρ(ί,τ) dt' m+l dx^O

(1.30)

(1.31)

THE DIRECT PRODUCT OF DISTRIBUTIONS 35

uniformly as Δί —♦ 0 for all τ lying in any compact subset of U. This proves that

• O a s A i - » 0 φ(ί + At) - ψ(ΐ) I d

Tt - ( « ( T ) , - r f r . T )

Put differently,

Ψ'(0=(βίτ),^(Λτ)

By using a similar technique and the method of induction, we can show that

ψ"" ' ( ί )=^τ) ,^( / , τ )

This proves that ψ (0 E D, and that the functional defined by (1.30) is meaningful. We now show that the functional / X g defined by (1.30) is also continuous on D,,T. The linearity is, however, trivial.

Assume that <p„(/, τ) —► 0 in D, T as v —> °°. Define

ΨΛ0 = <g(T), <pv{t, T))

Our objective is to show that φν(ί) —► 0 as v —* °° in D,:

Ψ?)(0 = <«(τ),φ?)(ί.τ)>

Here ^jfty, τ) stands for ^ir(i , τ). If i/>v(?) does not go to zero in £>,, then for some fixed k and an e > 0 there exists a sequence {f„}™=1 such that

* (0 . ^φ»( ί . τ ) ^ e (1.32)

V y = 1,2,3,4 If <pv(t, τ) —► 0 as v —► °° in 23,,T, then there exists a compact set K of R2 containing the supports of all φ„(ί, τ) and

sup ί.τ Λ* φΛί, τ) 0 as v —► oo

Therefore sup, |τφ„(Λ τ) —» 0 as v —► oo uniformly V τ G U. Likewise ψ^\ίν) —» 0 as v —► oo. This contradicts (1.32).

Therefore / X g as defined by (1.30) E D,'T. We now prove that δ(ί) X δ(τ) = δ(/,τ):

<δ(ί) X β(τ).φ(ί.τ)> = <δ(ί),<δ(τ),<ρ(ί,τ)»

= <δ(/)(φ(ί,0)>

= φ(0,0)

= <δ(Γ.τ).φ(/.τ)>

36 SOME BACKGROUND

Therefore

δ(/) X δ(τ) = δ(ί, τ)

One can see that 6(f) X δ(τ) = δ(τ) X δ(ί). If/ and g are locally integrable functions and if / g are the corresponding regular distributions, then

( / X g, <p(t, T)> = / / ( f ) , j <p(t, T)g(r) dr\

= 1 1 <p(t,T)g(r)f(t)drdt J — 00 J—00

= / / <p(t,T)f(t)g(T)dtdT J — 00 J —00

The support of <p(f, τ) #(τ) /(f) is bounded, so the switch in the order of integration is justified. This shows that

f*g=gXf

It can be also shown that

δ(ί)Χ 1 + ( τ )= 1+(τ)Χδ(/)

In general, we have / X g = g X / . This result is true in view of the fact that the space of testing functions of the form

<P(t, τ) = Σ ΨΛΟφΛτ) V

(where ψ„(ί) G V, and <ρ„(τ) G 2?T and the summation has a finite number of terms) is dense in D,T. See Zemanian [109, pp. 119-120].

1.9. THE CONVOLUTION OF FUNCTIONS

The convolution h(t) of two function f(t) and g(t) defined on the real line is given by

Kt) = if * g)(t) = / /(T)g(f - j)dr (1.33) 7 - 0 0

provided that the integral exists. We want to extend the definition of convolution to distributions so that the definition (1.33) may be true for the convolution of regular distributions.

If ψ G D, then

(h, φ) = (f * g, φ)

/

OO ΛΟΟ

<P(0 / f(r)g(t - r)dTdt 00 J - 0 0

THE CONVOLUTION OF FUNCTIONS 37

Use T = x, t = x + y,

(f*g,<p)= í Í Ax)g(y)<p(x + y)dxdy (1.34) J — 00 J — 00

The form (1.34) resembles the definition of the direct product of regular distributions. Therefore we should define the convolution / * g of two distributions /, g by

</*#. = </(')Xg(T),«p(/ + T)>

= (f(t),(g(T),<p(t + r))) (1.35)

But a problem arises in the definition (1.35) in that the support of <p(f + τ) is not bounded. In fact, if the support of <p(t) is contained in the closed interval [a, b], then the support of φ(ί + τ) should be in [(/, τ ) : β £ ΐ + τ ί / ) ] , which is unbounded (see Figure 1.1). Now let Ω = [support of / X g] Π support of <ρ(ί + τ) and let λ(ί, τ) be an element of T>, T such that λ = 1 over an open set containing Ω. We can now give a meaning to the convolution f * g whenever Ω is bounded by

</ * g· <P) = </(0 X g(T), λ(/, τ)φ(ί + τ)> (1.36)

This replacement is legitimate, since the value of a distribution depends upon the value of φ in a neighborhood of the support of a distribution and is not altered by changing the value of φ(ί, τ) outside a neighborhood of the support of / X g. (See Theorem 19.)

s \

Figure 1.1

38 SOME BACKGROUND

Theorem 21. Let / e T>'. A sufficient condition for the validity of the representation (1.36) for the convolution f * g asa distribution on IR of two distributions / and g over IR is that

Ω = [support / X g] (Ί support φ(ί + τ)

is bounded. This is ensured when one of the following conditions hold:

i. Either / or g has a bounded support. ii. Both / and g have supports bounded to the left; namely there exists a constant

T] such that f(t), g(t) are both zero for t < T\. iii. Both / and g have supports bounded to the right; namely there exists a constant

7"2 such that /(f) and g(t) both are zero for t > 7*2.

The proof is easy and is left as an exercise to the readers. The interested reader may consult Zemanian [109, pp. 123-124].

Examples. Show that

δ * / = / (1.37)

δ«"» * / =/<"·> (1.38)

We know that

( / * δ, φ) = (/(i), <δ(τ), λ(ί, τ)<ρ(ί + τ)>)

= </(/), λ(/,ο)<ρω> = </ω.<Ρ(ο> {f * δ<*\ φ) = </(/), <δ(*>(τ), λ(ί, τ)φ{ί + τ)>)

= (/(f),<8(T),(-l)*D*(A(f,T)(p(f + Τ))>)

= </(0,(-l)VWWA(0,T)>

= </(0.(-l)VW(0> = (0*/.φ>·

So if P{D) is a differentiation operator

a0 + axD + a2D2 + · · · + a„D"

then

P(D)f = (α0δ + α2δ" + · · · + αηδ(η)) * /

= [P(D)S]*f

THE CONTINUITY OF THE CONVOLUTION PROCESS 39

It can be easily shown that

IT (f * g) = f* * gw

where p + q = m,p,q = 0,1,2, In a special case

Om(/ * g) = /<m) * * = / * g(m)

1.10. REGULARIZATION OF DISTRIBUTIONS

Sometimes in distributional analysis we map a distribution into a C°° function by a convolution process that is called regularization of the distribution. We state without proof a theorem on regularization. The proof of this theorem can be found in any book on distribution theory.

Theorem 22. Let / G O ' and φ G Ί). Then h = / * φ is an ordinary function given by

h(t) = (f(x)Mt-x))

Also h{t) G C°, and

Α(*)(ί) = ( /ω .φ ( * ) ( / -*)>

See Zemanian [109, pp. 132-133].

1.11. THE CONTINUITY OF THE CONVOLUTION PROCESS

Let T)'R and T>'L be distribution spaces whose supports are bounded to the left and to the right, respectively. We say that a sequence {/}°1, of distributions in T>'R converges to a distribution / G T>'R if each of the elements f\, fa, f - ¡ , . . . are in T>'R and lim,_o f¡ = f in the weak distributional sense. Similarly we define the convergence in O'L. We now state the following theorem without proof:

Theorem 23. Let g be a given distribution, and let {/■} be a sequence of distributions tending to / weakly. Then the sequence f¡ * g converges / * g in T)' if one of the following conditions is satisfied:

i. g G £'. ii. {/„} converges to / in E'.

iii. {/„} converges in T)R and g is also in T>'R. •v- {fv} converges in D[ and g is also in T>'L.

40 SOME BACKGROUND

1.12. FOURIER TRANSFORMS AND TEMPERED DISTRIBUTIONS

1.12.1. The Testing Function Space S(W)

A C°° function φ defined on IR" belongs to S(W) if

sup |yV>( j t ) | < oo (1.39) Ms* x£R"

for each k, \m\ = 0,1,2,3 From (1.39) we can derive that

lim xmda<p(x) = 0 (1.40) M.|a|=0,1.2.3,...

if φ E 5(R"). Another way of describing conditions (1.39), (1.40) is

sup lim (1 + \x\2)rda<p(x) = 0 (1.41)

r, \a\ = 0,1,2,3 The set of conditions (1.39), (1.40), and (1.41) are equivalent conditions. As an example, e-t*i+*2+····*») g 5(R").

The topology of the testing function space S(W) is generated by the sequence of seminorms {γ^|„|}, k, \m\ — 0,1,2 where

γλ,Ι„,(φ) = sup I*"1 <?*>(*)!, φ e S(Rn) (1.42) jreR" |a |£*

it,|m| = 0,1,2,3 The topology generated by the separating collection of the sequence of seminorms

{7*.m}* |m|=o a s defined by (1.42) can also be generated by the sequence of seminorms {yrK°=0' where

7r(<p) = sup (1 + \χ\2Υ\δαφ\,φ Ε S(W) (1.43) xeR" lolsr

We can also generate this topology by the separating collection of seminorms {γ|α||„|} |er|,|wi| =0 ,1 ,2 ,3 where

->Wlm|(<P) = sup \xmda<p(x)\, φ e S(W) (1.44) ^eR"

Therefore a sequence <p„(jc) in S(W) —> 0 as v —» oo if and only if xmda(pv(x) —» 0 uniformly on W for each \m\, \a\ = 0,1,2,3 Consider

<pm(x) E S(R") and

A BOUNDEDNESS PROPERTY OF DISTRIBUTIONS 41

y\a\.\ß\(<P) = sup \xaSß<pm\ -> 0 as \m\ -> «>

Clearly <pm(x) -► 0, m -> oo, in the topology of S(R"). The function £-<Ι*ιΜχ2Ι U.I) diminishes to zero faster than any polynomial as

|JC| —* oo, but this function does not belong to 5(R"), since e - w is not differentiable at the origin.

1.13. THE SPACE OF DISTRIBUTIONS OF SLOW GROWTH S'(R")

A continuous linear functional over S(W) is called a distribution of slow growth. The space 2 W ) C S(M"), the identity map (embedding) IKW) -* S(W) is continuous, and THU") is dense in 5(R"). A distribution of slow growth can be identified one to one by an element of D'iW). Therefore

D'(R") D S'(W)

Every element of 2XR") and of S(U") can be identified as a regular distribution in S'(R"). Therefore

E'(W) C TKU") C 5(R") C S'(R") C T>'(W)

A distribution of slow growth is also called as a tempered distribution. 5'(R") is a proper subset of 2?'(Rn). The distribution ΣΓ=-» δ ( ' ~ " ^ e ^'C")

does not belong to S'(R). If φ E D, oo

(^δ(/-Λ)^2,φ(ί))= £>"2<P(«) n=-oo

So 5Ζ™=_οο δ(ί - n)e" exists as an element of D'. It does not belong to 5', for

( 00 \ 00

-oo / n=-oo

which diverges for φ(χ) = e_Jr2.

1.14. A BOUNDEDNESS PROPERTY OF DISTRIBUTIONS OF SLOW GROWTH AND ITS STRUCTURE FORMULA

Theorem 24. Let / e S'(R"), and let ψ G 5(R"). Then there exists a constant c > 0 and a nonnegative integer r such that

|</, «p>| < crr(<p) (1.45)

where γ,(φ) is the same as defined by (1.43).

42 SOME BACKGROUND

Proof. If (1.45) is not true, then for any integer m > 0 we can find a function <pm(x) ε S(W) such that

\{f, <Pm)\ > W7m(«Pm)

/. mym((pm) > 1 (1.46)

y¡

mym(<pm)J m

mym(<Pm)J

— ► 0 asm—»oo

as m —»oo

for each Í = 0 ,1 ,2 ,3 , . . . , which contradicts (1.46). Therefore (1.45) must be true. □

Since the sequence of seminorms {γΓ}"=0 is separating, it follows that the space S(W) is metrizable. Further it is a locally convex, sequentially complete, and Haus-dorff TVS, and therefore it is also a Fréchet space.

1.15. A CHARACTERIZATION FORMULA FOR TEMPERED DISTRIBUTIONS

Theorem 25. A distribution in W is tempered if and only if it is a finite sum of derivatives of continuous functions growing at infinity slower than some polynomial.

Proof. Sufficiency is obvious. So we need only to prove the necessity. From Theorem 23 it follows that if / G S'(U") and φ e S(W), there exists a

nonnegative integer r and C > 0 such that

K / » | < c ] r sup |(1 + W W Q ) W I

|</»|< Σ C' (1 + \x\2Y (fX φ(χ) ¿HR")

We have used the fact that

-00 J —00 J — 00

where

> = dx„ dxn-

\>(p(x)dx = φ(χ)

d d dX2 dX\

A CHARACTERIZATION FORMULA FOR TEMPERED DISTRIBUTIONS 43

Now let N be the number of «-tuples a such that

\a\ < n + r

We consider the product space L1 X · · ■ X ¿ ' = ( L ' f and the injection

ofSíR'JintoíL1)"· Using the Hahn-Banach theorem, we extend it as a continuous linear form in

the whole space (Ογ1. But the dual of (Ll)N is canonically isomorphic with (L°°)N'. Therefore there exist N V° functions ha, \a\ s n + r such that

/ = Σ a"m + U\2Y(-i)Mha] |a|Sn + r

Set

«aW = / . . . / ha(tut2,...tn)dh Jo Jo

\ga(x)\^\xx\\x2\--\Xn\\hay

Furthermore we have

ha(xi,x2,...,x„) = >ga

Therefore we have

/ = Σ *a[o + \x\2Y{-ir>ga] \a\^r+2n

■ dt„

a\*r+3n L \ I T \X\ /

Pa(x,t) is a polynomial in JCI,JC2 xn and t depending upon a. D

Theorem 26. Let J be the Fourier transformation operator defined by

Ja (7φ)(χ)= / <p(t)e"xdt

Then

J : S(U") -> S(W)

44 SOME BACKGROUND

is a homeomorphism and so also the operator J"-1 defined by

Proof. Let φ Ε 5(R"). Then

(J<p)(i)W = / <P(O0OV'*A

Using Theorems 4 and 8, we get

(ix)a(^)(k\x) = / (ixYrtrntfj'-'dt

= f (-Dr[<p(t)(it)k]ei,xdt. JR"

Therefore \χα&Τφ\ < °°. From this result it follows that if φ E S, then Τφ G S, and if <pv —> 0 in S, then (!F<pv) —► 0 in S. This indicates that the operator J : S —» S is continuous. It is also one to one as J φ = 0. J 7 - 1 exists and is given by

(277)" L (2TT)" 7 R .

Now we can show similarly that J7 - 1 : S —» S is continuous. Since J", J - - 1 are both continuous, J 7 : 5 onto itself is a homeomorphism. D

1.16. FOURIER TRANSFORM OF TEMPERED DISTRIBUTIONS

Let / e S'(W) then its Fourier transform Jf is defined by

(If,<p) = (f,?<p) V < p e S (1.47)

this is an analogue of the classical Parseval's relation, namely

/ (Jf)M<p(x)dx = / f{x)(J<p){x)dx V<p(ES (1.48) J— 00 J —00

if/ is absolutely integrable. So the classical Parseval relation (1.48) may be encom-passed by the definition (1.47) as a special case. There are other types of Parseval relations such as

/ (Jf)(w)(J<p)(w)dw=2n f f(t)(p(-t)dt (1.49) J — 00 J— 00

But the majority of mathematicians use (1.49) to extend the Fourier transform to distributions, and we will stay with the majority.

FOURIER TRANSFORM OF TEMPERED DISTRIBUTIONS 45

The definition (1.47) says that the Fourier transform of a tempered distribution is a tempered distribution, and it assigns the same number to φ G. S(W) as / assigns to (JV)(w). We can also define the inverse Fourier transformation operator by

(T-lf,f>) = {f,T-\) (1-50)

for all / E S'(R"). It is easy now to see that

II~l = T~XT = / so

J : S'(W) — S'(Un)

is also a homeomorphism (in the weak topology on S'(R")).

Examples. Consider the Fourier transform of the δ function and 1:

<J8,<p) = <8, J<p) = / « ( * ) . / <p{t)ei,xdx

= / <P«)dt J-00

(Γι8.9) = ( s . ¿ f <p(t)e-"*dt\

Therefore

or

T

- L ¿φ<,)<" --¿

'(¿)-Jl = 2πδ

fS = 1 J81 = (-ix)

JS(k) = (-ix)k

and so on, can be easily verified. Note that the derivative of a tempered distribution / is also defined by

<<?"/» = </.(-l)W<?a

46 SOME BACKGROUND

Theorem 27.* Let / e E'. That is, let / be a distribution of compact support. Then the Fourier transform of / is given by

( J / ) W = </(*). «"■'>

We will need to prove four lemmas before we can prove the main theorem.

Lemma 2. Let / £ £ ' and F(x) = (f(t), e>'x). Then F(x) is infinitely differentiable, and £<*>(*) = (/(/). (//)*<?''">.

Let us prove this lemma for k = 1. The general result can be proved by using similar techniques and the method of induction.

F{x + Ax) - F(x) I ■' fX+*x

-\j(i),ue· ) = t j\i)

rχ+^χ

Αχ <f(t),itei,x) = /f(t), £ j f [e"< - eitx}dz\ (1.51)

= JL Γ 0 ( Δ * ) = ^ / [ί?"ζ-ίτ"*]</ζ

/•χ+Δχ [¿*(ίζγ - ¿"(ixftdz

¡t rx+Δχ — / [ « " W 1 - ( « ) * " I « t o ] < k

—► 0 or Δχ —► 0 uniformly for t lying in any compact subset of R. Therefore θ(Δχ) —► 0 in E as Δ* —> 0. Hence letting Δ* —> 0 in (1.51), we get

Similarly, by induction, we can show that

F(k\x) = (f(t),(it )V">

Lemma 3. The integral

j <f(x),eix,)<p(t)dt = íf(x), J <p(t)eix<dt

We can partition the interval [~N,N] to define the integral over [— N,N]. Then

I {f(x),eia)<p(t)dt = lim ¿ ( / ( a t ) , ί"*>φ</,)Δ4 J-N ||Δ||—0^-'

♦Theorem 27 is also trae when / £ E\W). The proof is similar.

FOURIER TRANSFORM OF TEMPERED DISTRIBUTIONS 47

lim (/(*). Υ>ίχ,ί<ρ(ί,)ΔΛ ill—0 \ ¿-^ I ΙΙΔΙΙ

rN

uniformly over any compact subset of the real line (for x) due to uniform continuity of functions involved. The derivative with respect to x of the foregoing summation will also tend to zero uniformly with respect to x, lying on any compact subset of the real line. Therefore

J (fix).eu')<pit)dt = (f(x)f_ <Pit)eix'dt\

Lemma 4. Let φ G S and N EU. Then

f{x), j <pit)eix'dt\ — 0 as N -»°°

also

( / ( * ) . / <pit)eix,dt\^0 asN

We will sketch the proof of only one of these two, since the other one can be proved similarly.

Let

Φ(χ)

ip{k)ix) = / <pit)iit)keij"dt

f <pit)eix,dt JN

f JN

\φ(ί\χ)\ < / |ί|*|φ(ί)|Λ -» 0 asN — ι JN

uniformly for all x £ R. This completes the proof of Lemma 3.

Lemma 5. Let / e S' and φ £ S. Then

J_ <f«).e>")<p(t)dt = (f(x), J_ eix'<pit)dt

Proof. By Lemma 3

J {fix),eix,)<pit)dt = (fix), J eix'<pit)dt\

48 SOME BACKGROUND

Now let N —> «:

= (f(x),j eix,<p{t)dt + J eixt<p{t)dt + j eix'<p(t)dt\

J ( / ( χ ) , £ ώ Μ ί ) Λ = / / ( χ ) , / eix'<p{t)dt

Proof of Theorem 27. Let

W,<p) = (f.I<p)

Therefore

fix), j <?(ty"dt J— oo

= I (f(x),ei,x)<p(t)dt J -an

= «/(*U"*).<P(0)

(J/)(0 = </(*). «"*>

Case 1.

J8 ( i ) ( 0 = (6(i)(i).e"x)

= <6(r),(-i)*(/*)V">

(J6 ( i ) )W = ( -« )*

Case 2. If / and g are two distributions with compact supports, then their con-volution is also of compact support and their Fourier transform is given by

W ' * S)(w) = <(/■* g){t),eiw·)

= (f(t),{g(r),e^'+T)))

= {fi.t),eiw,)ig(r),e^)

= (Jf)(w)(Jg)(w)

FOURIER TRANSFORM OF DISTRIBUTIONS IN 27 (R") 49

1.17. FOURIER TRANSFORM OF DISTRIBUTIONS IN &(Un)

We will prove our theorem for T>' = T)'(U') (a method of Leon Ehrenpreis [36]) and the general results for T>'(W) are similar. Our main objective will be to prove that the space Z of testing function space, which is entire and satisfies the condition

|ζ*Φ(ζ)| < ckeaM, k = 0,1,2,3 V<DGZ (1.52)

is the class of those entire functions that are the Fourier transforms of φ G T> whose support is contained in the interval [—a, a].

Theorem 28. The Fourier transform of an element φ G T> with support in [—a, a] can be extended as an entire function Φ(ζ) such that

|ζ*Φ(ζ)| < cke"M for each k = 0,1,2, . . .

Conversely, if there is class of entire functions Φ such that

|ζ*Φ(ζ)| < cke"M (1.53)

then the inverse Fourier transform <p(t) of Φ(ζ) defined by

is an element of Ό with support contained in the interval [ —a, a].

Proof. Let <p(t) G T> such that the support of <p(f) is contained in the interval [—a, a). Define Φ(ζ),

Φ(ζ) = / <p(t)e"zdt

Clearly Φ(ζ) is the extension of the Fourier transform of <p(t) G D as an entire function. Now

ζ*Φ(ζ) = [ <p(t)zkei,zdt = ik [ φ(*>(0<;"ζΛ J— 00 J— 00

|ζ*Φ(ζ)| < eaM i \<plk)(t)\dt, z = x + iy (1.54) J -a

|ζ*Φ(ζ)| < ckeaM

where

c*= f Wk\t)\dt J —a

50 SOME BACKGROUND

Conversely, let Φ(ζ) £ Z such that (1.53) is satisfied. Denote the inverse Fourier transform of Φ(ζ) by <p(t). Then we have

<p(t) = - i - / Φ(κ>)ίΓ'»"<Λν and <p(/) e c"(R)

1 f" . = — / Φ(»ν + ι » ^ ,<K,+0,)'<ÍH> (by contour integration)

tt J-oo = J _ Γ Φ(νν + ζ»(νν + / ν ) 2 ^ , ( . + , , ^ Η

27Γ y.«, (w + i » 2

•«01 * ψ- Γ -^V"1 2π y_„ w2 + y2

~ 2\y\e

For y > 0,

as y —> oo when t < —a. For y < 0,

ItfOl ^ ^ ( f l + , ) - 0

ItfOl < ^|^('"α ) - 0

as y —* —oo when / > a. Therefore <p(f) = 0 for \t\ > a. D

Our mission is not fulfilled as yet. We want the Fourier transform to be a home-omorphism from D onto Z. To achieve this objective, we have to transport the topology of the space T> onto Z by means of Fourier transform operator F and define convergence in the space Z accordingly.

We look into the inequality (1.53) and note the fact that, when φν —* 0 in Ό, there exists a compact subset, say, the interval [—a, a] containing the support of each <p„(f). Therefore we define that a sequence {Φ„(ζ)}"=1 in Z converges to zero in Z as v —► °° if and only if

1. Each Φ„(ζ) £ Z. 2. There exist constants c* > 0 independent of v such that

|ζ*Φ„(ζ)| =£ cke"w, ¿ = 0,1,2, . . .

3. {Φ„(ζ)}"=) converges uniformly on every bounded domain of the z-plane.

It is now a simple exercise to show that

<pv(t) - ^ O i n B * * Φν(ζ) — 0inZ; <pv(t) = (^_1Φ)(/)

EXERCISES 51

With this definition of convergence in Z, the Fourier transform becomes a homeo-morphism for Ί) onto Z. The space Z' is called the space of ultradistributions.

We have proved the one-dimensional case only. The general case can be cited and proved analogously. It is left as an exercise. So we define the Fourier transform of φ £ T>'(W) by the relation

<J/. = < / . J » ν φ ε ζ ( 0 Τ )

Note that Jf is an ultradistribution that assigns the same number to φ £ Z as / assigns to J<p. The generalized function

00

0

but it does not belong to S'. We can find its Fourier transform as follows: Let

ao

&f.9)=(f.f_ <pix)ei,xdx\

r°°

Therefore

00

inx

:F/ = £ V V n=0

is an ultradistribution.

EXERCISES

1. Prove that in a topological vector space E over the field of complex numbers, a set different from 0 and E cannot be both open and closed.

2. Let £ be a vector space, and let U be a subset of E that is convex, balanced, and absorbing. Prove that the sets ^U (n = 1,2,...) form a neighborhood base at zero in a topology of E that is compatible with the linear structure of E. What happens when you drop the assumption that U is convex?

52 SOME BACKGROUND

3. Prove that the product of a family of topological vector spaces Ea, a G. A is Hausdorff if and only if every Ea is Hausdorff.

4. Prove that a TVS E is compact if and only if it consists of a single element 0. 5. The convex balanced hull of a subset A of a vector space E is the smallest

balanced convex set containing A. Give an example of a closed subset of the plane R2 whose convex hull is not closed.

6. Prove that the Schwartz testing function space T)(U") is not metrizable. 7. Let {£„}„= i be a sequence of Fréchet spaces. Prove that the product space E =

ΙΊ/L i £; is a Fréchet space. 8. Let K be a compact subset of R", and let C^(K) be the space of complex valued

functions defined on R" having supports in K. Define the topology on C£(Af) whose neighborhood basis about the zero element is

7(i«p e) = ( 0 vary in all possible ways. Prove that C^(K) is a Fréchet space.

9. Let G»(R") be the space of continuous function in W that converges to zero at infinity, equipped with the topology of uniform convergence on W:

φ -► sup |<p(*)|

Prove that Coo(R") is a Banach space. 10. Give an example of a continuous function / in R" with the following two

properties: (a) there is no polynomial P in W such that

|/(JC)| s \P(X)\ VxGW

(b) The distribution φ -* J <p(x)f(x) dx is tempered. 11. Is the function e^ Fourier transformable in the distributional sense? (Hint: De-

termine if e'*' is an ultradistribution. Find the Fourier transform of ¿''*'.) 12. Compute the Fourier transform of the Heavyside unit function

t > 0 t < 0

13. I f / , geS(R" ) , define

"(')={ό:

Find the Fourier transform of ( / * g)(x).

EXERCISES 53

14. By using the Paley-Wiener theorem, or otherwise, show that if P (^ ) is a differential operator with constant coefficients (not all zero) in U". The equation P ( ¿ ) u = 0 has only one solution in the space E' that is u = 0.

Note. Readers who have a strong background in functional analysis may want to use Treves [101] and [103] to solve these problems. These are the prime sources.

2 THE RIEMANN-HILBERT PROBLEM

2.1. SOME COROLLARIES ON CAUCHY INTEGRALS

The first four sections of this chapter briefly present some of very useful results given by Muskhelishvilli [65] and Carrier, Krook, and Pearson [17] for Cauchy integrals. Let F(z) be defined by

TO.'/fle* ,2.1, 2m Jc t-z

where C is some curve in the complex /-plane, /( /) is a complex-valued function prescribed on C, and z is a point not on C. The curve C may be an arc or a closed contour, or more generally a collection of such arcs and closed contours. For suitable curves C and functions / , F(z) will be analytic function of z. For example, if f(t) is continuous on C and C is a smooth curve, then F(z) will be analytic. However, the conditions on C and /( /) can be further relaxed and still yield the analyticity of F(z). If C is a closed contour, the positive direction will be counterclockwise and F+(fo) and F-(fo) where t0 £ C are limits of F(z) as z —♦ t0 from inside and outside of C, respectively. If C is a curve extending from point A to point B and if we orient ourselves at point t0 on C so as to be facing in the positive direction of integration from point A to point B, then the limits of F(z) as z —► f0 from the left and from the right, if limits exist, will be denoted by F+(t0) and F-(io), respectively. We define the principal value Fp(to) of the integral (2.1) by

2m t-*oyc^c€ t - i0

54



SOME COROLLARIES ON CAUCHY INTEGRALS ss

where Ce is the portion of the curve C contained within a small circle of radius e, centered on fo· Our objective is to find a relation between FP(ÍQ), F+(to), and F_(/o). To derive such a relation, let us first assume that /(f) is analytic at point t0. That is to say, there exists a circle ξ with center /0 and radius r > 0 such that /(/) is analytic in the disc {f: \t - i0| < r}.

Let z be a point in the circle situated to the left of the curve C. Note that

/ í^-dt = / í^-dt (Cauchy's theorem)

where C\ is the arc of a circle with center t0 and radius e < r, as shown in Figure 2.1.

F(z) = — / ψ— + — / ψ— (2.2) 2-τπ yc_c< t-z 2m JC] t - z

Therefore letting z —»r0+ in (2.2), we have

2m yc_c< ί - ίο 27Π yC| f - f0

Now let e —» 0, and (2.3) becomes

F+(t0) = F„(f0) + \f(t0)

Similarly, by changing the sides of z and C\, we get

F-ih) = FpUo) - \fdo)

(2.3)

(2.4)

(2.5)

Figure 2.1

The relations (2.4) and (2.5) are known as Plemelj formulas. It is a simple exercise to show relations (2.4) and (2.5) by assuming only that / ( / ) is continuous on C and satisfies the Lipschitz condition

| / ( f . ) - / ( f o ) | < A | i . - ' o P (2.6)

for all points t\ on C and outside C in some neighborhood of to, where A and a are constants, with 0 < a ^ 1. We are now assuming that /(f) satisfies (2.6) in a neighborhood of t0, containing the curves Cf and C\.

Let

F(z) = - L f Δ!λώ (2.7) 2m Jc t - z

and Ce be an arc of C lying in a neighborhood of the point to satisfying (2.6). Then

2™ Jc-c. t-z 2m ic.t-z

2m Jc-Ct t - z 2m JCt t - z 2m JCi t - z

= -L/ ^ ^ + - L / / ( f ) - / ( z ) ^ + ^/-L^(2.8) 2πΐ y c - c , ' _ z 2τπ 7C< / - z 2m Jc, t - z

where Ci is an arc of a circle lying on the side of the curve C opposite to the side point z lies on, such that the end points of C\ lie in the region of C where the Lipschitz condition holds true.

Letting z —> r0 from the left and then e —> 0, in (2.8) we get

F+(h) = Fpdo) + z/(fo)

Note that by Cauchy's theorem Jc y^dt = fc ~r2 dt —> m and that the second integral in (2.8) —» 0 as z —»f0 from the left and then e - » 0 (see Example 1).

Similarly we can prove that

F-(to) = Fp(t0) - l-f(t0)

This shows that to prove (2.4) and (2.5) the condition of analyticity at fo can be replaced by a Lipschitz condition in a neighborhood of the point to- A stronger result is given in Section 2.2.

2.2. RIEMANN'S PROBLEM

In his Ph.D. dissertation Riemann considered the problem of determining a function W+(z) = u(x,y) + iv(x,y) that is analytic inside a closed contour C such that the boundary values of its real and imaginary parts on the contour C satisfy the linear

RIEMANN'S PROBLEM 57

relation

α(/)κ(ί) + ß(t)v(t) = γ(/) V t on C (2.9)

where a(i), /3(f), γ(0 are given real functions. In view of the Riemann mapping theorem concerning conformal mapping the closed contour C can be taken as the unit circle. We define an analytic function W_(z) outside the circle C by the relation

(0 W-(z) = W+ ( - ] (2.10)

For z on C we have

That is, z = \ V z on C. Therefore z —► / implies that \

zz

->

W-

= |z|:

/ and

(0 =

2 _

w,

1

(0

or

or

Since

we have

W-(t) = W+(t) = u(t) + iv(t)

W-(t) = u(t) - iv(t)

W+(t) = u(t) + iv(t)

a ( 0 , „M, ) ,«M0 ( 2 | 1 )

and

_ WAO-WA') n ... ι/(ί) = (2.12)

2/ Substituting for u(t), v(t) from (2.11) and (2.12) in (2.9), we get

^^w^^^iimw^^^ (2,3)

The Riemann problem is now reduced to finding functions W+(z), W_(z) analytic inside and outside the circle C, respectively, such that their boundary values on the circle satisfies the linear relation (2.13). The behavior of W-(z) at <» must also be prescribed for completeness. From the relation (2.10) it follows that W- (z) —► W+ (0) as z —* oo.


2.2.1. The Hubert Problem

Hubert generalized the Riemann problem posed above by determining a function W(z) analytic for all values of z not lying on the curve C such that for t on C

W+(t) = g(tW-(t) + fit) (2.14)

where /(f) and g(t) are given complex-valued functions (or even generalized functions as considered in this book) and W+ (/) and W- (t) are limits of W(z) as z —> / from inside C and outside C, respectively.

2.2.2. Riemann-Hilbert Problem

Hubert in his problem described above considered only the case of a closed curve or circle C. But a more general case where C is an arc, a closed contour, or a collection of arcs and contours has come to be called the Riemann-Hilbert problem. The behavior of W(z) at oo must be specified, as must also be the behavior of W(z) near the ends of C if C is an arc. If C is an arc extending from point A to point B and the positive direction on the arc C is considered as the direction pointed by an arrow sign pointed toward B, then W+(t) is interpreted as the limit of W(z) as z approaches t from the left of C and W~ (f) is interpreted as the limit of W(z) as z approaches t from the right ofC.

2.3. CARLEMAN'S APPROACH TO SOLVING THE RIEMANN-HILBERT PROBLEM

Carleman encountered the Riemann-Hilbert problem in his work on singular integral equations and devised a very effective means of solving it. The method is first to find a nonzero holomorphic function L(z) that is analytic everywhere on the z plane except possibly on the curve C, satisfying

L+(f) = g(t)L-(t) (2.15)

and such fhatL+(f),¿-(0 are nonzero. Substituting (2.15) in (2.14), we get

HMO W-W _ /( ') L+(t) L-(t) L+(0 (2.16)

Since L(z) Φ 0, the function ^ is analytic for z not on C such that (2.15) is satisfied. Since L(z) is known, W(z) is known. If we take

Jc t-z « , * _ ! //W/ZMO Λ/(Ζ) = — ;

2m j c then using the discontinuity theorem we have

r ΛΜ0-ΛΜ0

CARLEMAN'S APPROACH TO SOLVING THE RIEMANN-HILBERT PROBLEM 59

Therefore the equation (2.16) becomes

W-(l) -M+(t)= - ~ - M _ ( f ) W+(0 L+(0 " ' " " ' L-(t)

and so the function jj4 — M(z) is continuous on C and in view of Morera's theorem is entire. Now the behavior of W(z) at °° is prescribed to be of polynomial order if we can select a polynomial pm(z) such that the solution W(z) of the equation jQ - M{z) = pm(z) satisfies the order condition W(z) = o(z"), z —> °o.

We now proceed to find L(z). From (2.15) we have

l n L + ( f ) - l n L - ( 0 = ln(s(0) (2.17)

We can solve (2.17) at least for the case where C is an arc, since we do not have to worry about the multiplicities of values of In g(t). We also assume that g(z) is continuous at end points zx and z-¿ of the arc having the values g(zx) and g(zi) at their end points. Using the discontinuity theorem, we then get

1 f ln(s(0)

which, for example, is Q(z). Thus

2m Jc t-z

L(z) = εΰω Ψ 0

Therefore

L-(f) = eQ~w

and

L-Ct) - elQ*W-Q-(D) - eln(«(0)

g(í)

We have now solved (2.15). We know that ta, 0 > a > - 1 , and that In t are integrable in the neighborhood of

the origin, even though these functions blow up at the origin. Such functions are said to have integrable singularities at the origin. Now, if C is an arc, we permit/(/) to have integrable singularities at the end points. If f(t) has an integrable singularity at an end point /„, then F(z) can grow no faster than some power ß > — 1 of \z — te\. In solving some singular integral equations of physical problems, such types of functions do arise.

Now there are two important cases to be considered: (1) L(z) is discontinuous at the end points z\ and z2 of the arc C and (2) when C is a circle, lng(f) may have multiplicities of values on C. These two cases are resolved quite easily as follows:

Case 1. C is an arc with beginning end point Z\ and the final end point z2 with a singularity at the point zx. The case where L(z) has a point of discontinuity at z2 can be resolved similarly. Assuming zi to be a point of discontinuity, we get

= _L /"lute: 2m Jc t

ß(z) - ■_ > ]n^t))dt

lc

- ^ lnCgCzOWz. -z ) Ζ7Π

or ¿ ( z ) ~ ( z - z , r 1 / 2 m l n ( g ( z , ) )

~ ( z - z , ) a + w (a, ¿being real)

L(z)(z-zx)k> ~ ( z - z , ) » + * ' + *

We choose k\ such that — 1 < a + k\ < 0. Thus the singularities at end point are taken care of in previous section, since L(z)(z — zi)*' has an integrable singularity at z = z\. All that we have to do is to replace L(z) by£(z)=L(z)(z- Z l )* ' .

Case 2. Now we resolve the case where g(t) has singularities at / = zx and C is a circle of unit radius. So In g(t) will change values by a multiple of 2m. We now can avoid the multiplicity of values as follows: Define

g0(t) = (t- z0yg(t)

where ZQ is a point within Co. Now define

for z inside C \u- z0yi N(z) , , « L ( 2 ) f o r z o u t s i d e c

Our problem is now resolved by finding the solution of

N+(t) = goW-0)

where g0(t) is a single-valued function on C. The same procedure as was used for the arc may now be applied. Having determined N(z), we get L(z) as follows:

( N(z) for z inside C

¡m? for z outside C

We now solve a few related problems:

Example 1. Let / ( / ) be continuous on a smooth and rectifiable curve C and satisfy \f{h)-f{t0)\<Mu-to\a

for all points ii on C and outside C in some neighborhood of t$ E. C where A and a are constants with 0 < a £ 1. Show that Fp(t0) exists and the Plemelj formulas hold.

Solution. Refer to Figure 2.1, and set

TO_ · /£«* 2m Jc t-z

2m Jc-Ct t-z 2m JCt t-z 2m Jc< t - z

In view of Cauchy's theorem, we have

2m Jc,1 - z 2m Jc, t - z

Now let z —► ίο + · Therefore

2m 7C_C < t - t0 2m *-«o J C , t - Z

The proof for the existence of Fp(to) is sketched in the next example. Letting e —> 0, we get

F+(/0) = Fp(i0) + — 7 - + hm hm —- / — dt

2 e-»0z-io 2m JCt I — z

Now we show that the last double limit in the above equation is zero.

\t-z\a\dt\ 12m yc< f - z 2m Jc,

Denote / c Jfjrjf |df | by / and put t - z = pe^, where p = \t-z\ and φ = arg(r - z) = the angle between line zf and the line zt0. Therefore

dt _ dp + id<p t — z p

\dt\ =]dt\^\dp\ \t~z\ p p \d<p\

Hence

« ΙΛΙ \t~z\

= [ [pr-l\dp\ + p?\d<d] Jc,

With z being fixed, we can split up the arc Cf into subarcs (f0,t¡), (ti,t2)< (Í2,Í3) U„-i,t„) such that in each of these subintervals p is monotonic on C. For the sake of definiteness, we assume that p is monotonic* increasing for t between f,_1 and /, then

/y-w]-/y-*-? Ί

' / - I

-¿Of-rf-.] Thus we see that

I < ±[p<; +2p»2 +2ρ<ϊ + ■ ■ ■ + p°n] + ρ»Φ

where Φ is the total variation of the angle φ and p¡ is the maximum of all ρ,-'s, as discussed above. Clearly Φ ^ 2·7Ϊ\ Now z — t = z — t0 + t0 — t. Therefore \z - t\ < \z - i0| + e, and

ΉπΤ/< 1-\2{η- 1)6° + €α2ττ1 z—fo 2 l J

Letting e —* 0, we have

lim lim / = 0 £-.Oz->lo

Note that a geometrical interpretation of Φ can be that it represents the angle sub-tended by the arc Ce at z. We can assume that the curve C has a corner at point t0, making an angle Θ at t0 as shown in Figure 2.2. Assume also that the function /( /) satisfies the H condition in a small neighborhood of the point to as stated in (2.6).

Our Plemelj formula under the same set of conditions takes the form

F+Uo) = f1 - ¿ ) /Co) + Wh)

2TT

From (2.18) it follows that

F-(to)=-^zf(to) + Fp(t0) (2.18)

/(lb) = F+(i0) - F-(t0)

FP(to) = \ [F+(i0) + F_(io)] - Q - ¿ ) /Co) (2-19)

The proofs for these formulas become simpler if the H condition around to is replaced by analyticity of / at to. In that case we can also say that /(f) is analytic in a

*It is assumed that the arc Ct has at the most n oscillations, n being finite and a 0. We say that the curve C oscillates at point r if its slope changes sign at t.

Figure 2.2

neighborhood of f0, and we can replace 5J5 / c frj dt by ¿ / c ¿ ^ ^ with appropriate choice of Ce and C\.

We now give a much stronger result proved by Muskhelishvili [65, p. 38].

Example 2. Let / ( / ) be a complex-valued function continuous on a smooth curve C satisfying H condition:

1/(0 - /('o)l =s ¿I' - 'οΓ

for all points f on C in some neighborhood of to £ C where A and a are constants with 0 < a < 1. Show that Fp(t0) exists and the Plemelj formula holds.

Solution. By writing /(f) = /( /) — /(fo) + /(fo) and performing calculation as in the foregoing theorem, we get

F+(to) = \m) + Fp(to) + Hm lim ~ f fit)~fi'o)dt

2 €—Οζ->(ο 27Π Je, t — Z

We can assume the existence of Fp(t0) for the time being:

F+(to) = ¿/Ob) + FpUo) + «-o 2τπ /c < f /Ob)

■io c/i

See Muskhelishvili [65, p. 38]. Let I' ~ h\ - r, φ = arg(f - t0) = the angle that the chord f0f makes with the

tangent vector to the curve C at fo· Then

f - f0 = re*

ln(f - f0) = In r + ΐφ

dt dr t - t0 r

+ Ίάφ

Let

Therefore

or

, i t m-m)dt 2™ Jc, ' - to

2ττ JC, U - t0\

«s=['.·*'?]*^· as fc \d<p\ ^ m. This is because for a smooth curve we have

dr . — — ± cos a ds

0 < a ' < o o < y [65, p. 10]

where a' is the acute angle that the tangent to C at t makes [65, pp. 10, 425] with the chord tot. The upper sign refers to the upper arc, and the lower sign refers to the lower arc. Therefore r is monotonic with s.

Let ry = |i0 - tf\ and r2 = |/o — 1 where te, t¡ are two end points of the curve C£. By construction (see Figure 2.1), r\ < e,r2 < e. Therefore

A A |/| < - — 2 e a + —me -> 0 as e -> 0

27τα Ί.-Π

One of the Plemelj formulas follows. The other Plemelj formula can be proved similarly. The existence of Fp(t0) = ¿¡(/ ' ) / c 0fdt can be proved by doing similar manipulations.

Example 3 (Inversion of the Cauchy's integral). Let C be a closed contour, and let /(r) be a function that is continuous on C and satisfies H condition on C in a neighborhood of point to G C. Let

,(,0) = 1XP) f p L d t ™ Jc l - h

To prove that

•m Jc ' ~ h let F(z) be defined as

Then

F(z) = —. at, z S C in Jc t - z

F+(to) - F-(t0) = f(to)

F+(/0) + F_(io) =-.(P) f -ßt-dt m Je t - t0

Now define a function G(z) as

F(z), z inside C " {

Then

G ( z ) Ί -F(z), z outside C

G+(/0) + G-(fo) = F+(to) - F-(to) = /(/0)

G+(i0) - G-(io) = F+(f0) + F-Uo) = - / τ ^ - Λ = g('o) WJ Jc t - to

G{z) = — / dt Je t - z 2m

Now we can write

Therefore

G+(r0) + G-(fo) = - ( />) / - ^ τ - Λ

/Ob) = - ( n / ^ - Λ m Je t - t0

Note that if / ( / ) satisfies the // condition in a neighborhood of point /<> belonging to C, then so does g(t). For proof, see Muskhelishvili [65].

The foregoing formulas are also true when C is the union of contours L^,Li, L2,L$ Lp where L\,L2,... ,LP are nonintersecting contours contained in L0 and S+ stands for connected region bounded by one or several smooth nonintersecting contours Lo,Li Lp. The region 5 - is the complement of 5+ U L. Here L = L0 U L, U L2 U · · · U Lp. See Figure 2.3.

If L0 exists, then S_ = Sf U S2 U · · · U S~ U SQ where Sr,S2~~,... ,S~ are the interiors of contours L\,L2,... -Lp, respectively, and SQ is the exterior of Lo· If L0 does not exist, then S- = S^ U S2 U · · · U S~ and S+ = complement of 5- U L\ U L2 U ■ · ■ U Lp. Obviously S+ in this case is unbounded. The above


Figure 2.3

inversion formula can be put in the form

™ Jc L7" Jc T-t J ' -*o

which is called the Poincaré-Bertrand formula.

2.4. THE HILBERT INVERSION FORMULA FOR PERIODIC FUNCTIONS

The Hubert transform (///)(θ) of a periodic function /( /) with period 2π is defined by

(Hf)(d) = ^(P) J" /(β - Ocot *- dt

provided the integral exists. With a little careful calculation it can be shown that

{Hfm = 2ΐτ{Ρ)Γ fit)COt (^~0 ώ

1 f^1* ί Θ — t \ dt

When the periodic function / is W integrable forp > 1 over the interval [ — ir, π], it can be shown that

H'f =~f + _L Γ f(t)dt a.e.

We give the most elementary proof of this theorem by using the Poincaré-Bertrand formula as follows:

THE HILBERT INVERSION FORMULA FOR PERIODIC FUNCTIONS 67

Let /(f) be a complex-valued function, satisfying H condition on the unit circle or be analytic on C. Then in view of the Poincare-Bertrand formula, we have

im Je*-*

g(x)

i(P)/J<íU-/W m Jc x- t

Let t = «?'e, x = é1*,

dt

Now put

Therefore

or

ie ied6 = -: η: - I cot %-—^dQ + l-d6

x - 1 e'* - e,e 2 2 2

■m -ÁP) I f(eie)

-ÁP) [ g(eie) ™ Jc

1 cot r^dd + ^d0 2 2 2

i cot ^r^-de + i d e 2 2 2

/(«") = Φ(0)

g{ew) = ψ(0)

Λ2ΙΓ

-ά ( Ρ ) Γ φ ( θ ) ο ο ι (^) /•2ir

</β

+ -2ir

1

/ yo

Φ(θ)άθ = φ(θ)

- Ρ) iJ W)C0X{^) άθ

+ 2ττ Jo φ(θ)άθ = Φ(θ)

ζ(//Φ)(β) + (/Φ)(θ) = φ(θ)

ΐ(Ηφ)(θ) + (1ψ)(θ) = Φ(θ)

(/φ)(β)= -L [*Φ(θ)άθ ¿π Jo

Eliminating ψ between (2.24) and (2.25), we get

iH[iH<p + Ιφ]θ + /[ί(//Φ) + /Φ](0) = Φ(θ)

(2.20)

(2.21)

(2.22)

(2.23)

(2.24)

(2.25)

Since ΙΗΦ = 0 and ΗΙφ = 0, we have

-Η2φ + /2Φ(β) = Φ(0)

or

/2 Η2Φ - (/Φ)θ = -Φ(ο ) 1 /*27Γ

(//2Φ)(Θ) = -Φ(0) + — / Φ(0)Λ0 2 τ

Ιί/ο2,ΓΦ(0)<ί0 = 0, we get

(Η2Φ)(Θ) = -Φ(0 )

The Hilbert transform of periodic functions / with period 2τ, as defined in Chapter 7, is given by

(///)(*) = ¿ ( P ) ^ f(x - f)cot ( ^ ) (2.26)

The inversion formula for this transform is

(//2/)W = -fix) + γΎ[ /W ώ i2·27)

The inversion formula (2.27) at point x for the Hilbert transform of periodic function / with period 2τ is valid under the assumption that / is continuous over the period [—ΊΤ,ΊΤ] and satisfies the H conditions in a neighborhood of the point x (see Exercise 1). Butzer and Nessel [12] show that this inversion formula is true for any function that is periodic with period 2ιτ and is V integrable over the interval [—tt, 7T] for p > 1. But their inversion formula can be easily generalized for arbitrary period 2T. By a careful calculation it can be shown that the definition (2.26) is also equivalent to the following two forms:

(///)(*) = γτ{Ρ) J7 /(Ocot ((X ~ τ ' ) ΐ Γ ) dt (2.28)

or

(///)(*) = ±(P) J T f(x - t)cot ( ^ ) dt (2.29)

Example 4. Show that there exists an analytic functions F(z) defined inside the unit circle whose limit as z —» Θ on the boundary of the circle is of the form /(Θ) + ig(6) where g(d) = (///)(Θ). In this sense / and g are conjugate to each other. Finally,

show that if / (θ) = Σ™[αη cos ηθ + b„ sin ηθ], then

g(6) = τ_\αη sin ηθ — b„ cos ηθ)

Note that the constant term in the Fourier series of /(Ö) is missing. This is because the Hubert transform of a constant (periodic function) is zero.

Solution. The required analytic function F(z) should satisfy the following properties:

1. F(z) is analytic on the unit disk z : \z\ s 1. 2. F(0) = 0.

Let us denote the unit circle \z\ = 1 by C. Then

/ -Jc t

F(z) =±fF-^ (2.30) 2

Again

Therefore

Í F(z)dz =

Jc z 2mF(0) = 0

r2ir / F(e''*V) / ^ ^ (2.31) 2 2TTÍ yc t -10

F+Uo) - F(t0) as F is analytic on C, the unit circle. Therefore from (2.30) we get

27Π F(,o) = i F ( , o ) + A.XP) [ m * Jc t ~ h

or

7Π 7 C / - t0

Let

/ = e"

to = e"

Therefore

F(eie) = ^ I F(e'*) n Jc

2m Jc

φ- Θ cot —¿— +1 dq>

Φ - 0 Fie'*) col?—-dip

F(eie) = /(//F)(0)

/(<**) + ig(eie) = /(///)(0) - (Η8)(θ)

Equating the real and imaginary parts, we set

g = Hf

f=~Hg

Again, if z = em = cos 0 + /sin 0, then z" = cosn0 + i'sin/ιθ. By the foregoing result, we have

If

//(cos«0) = sin«0

//(sin ηθ) — -cos ηθ

00

/ (0) = ]Π(α„ cos ηθ + b„ sin ηθ)

then formally

g(0) = ] T a„//(cos «0) + ¿„//(sin /i0) n = l

// is the operator of the Hubert transformation: oo

g(0) = y j an sin ηθ — bn cos ηθ

It will be shown in Chapter 7 that

He"" = - lim /> / - i — A


= sin nx — i cos nx

Therefore, equating the real and imaginary parts, we get

H(cosnt) = sinn*

//(sin nt) = —cosnx

Example 5. Solve the following integral equation for / :

the function h being continuous in the interval (—1,1). The function / is to be continuous in the interval (—1,1) and may have integrable singularities at t = ±1 .

F(z) = ^-f í^-dt, l m z # 0 2m

Solution. Take

f-ß F + M - F - ( * ) = /(*)

m y_, t - x

The given integral equation reduces to the form

m(F+(x) + F-(x)) =h(x)

F^(x) = (-l)F-(x)+^$, g(x)=-\ m

We find a nonzero function L(z), satisfying

L+(x)

Take

ln(*(/))A

- 1

. . . . i y Hgu)

5-\ t-z 2 \-(\+z)J V z + 1

-1

2iri

m-^

72 THE RIEM ANN-HILBERT PROBLEM

We now make an adjustment for L(z) by choosing

i * > - ' lz~l z - 1 V z + 1 ^ / ( z - l ) ( z + l )

This adjustment is done in order to let L(z) grow algebraically with index between — 1 and 0 as z —> ± 1. The given integral equation can be written

F+Or) F-(x) h(x) L+(x) L-(x) L+(x)m

F(z) 1 /" A(0 / : L(z) 2m y_i L+(t)m(t - z)

where £] is an arbitrary constant. Thus

F(z) = 0(1), z - > »

= 0 ( z ± l ) " 1 / 2 a s z - > ± l

which is what we wish to have.

dt + k\

2m 7_, L+(t)m(t - z)~ Fiz) = ^ / . (,T. Jt + *,L(z)

Now

/ ( j r ) = F + ( * ) - f _ ( j t )

AW = *, [L+(x) - L_(JC)] + [L+00 + L_(JC)] Z.+(jr)2m

1 /"'

+ [L+(JC)-L-W]—(ny A(0

., L+(/)(r-x)iri

Uz) = -1

sJz^\s/7T\ L+(0 = Z^ L-(t) = ' -

y/l -t2 v/l -t2

Therefore

*.2< _ 2/ 1 /" AW y/\ - t2 y/\ - t2 2™ J-i L+(00 - X)m

2/*, 1 , „ , / · ' A(r) f2 7-1 £+( \ / l - ί 2 π \ Λ - f2 ■/-. ¿ + ( 0 ( i - Jt)in

dt


\ / l - r 2 77v/l +t2J~i (t-x)TrP

\ / l - r 2 ^ 2 Vl - 1 2 J-i ('-*)

= Λ + i (f) /"' JEl^lÉL

v/ i - f 2 ττ2ν/ΓΓί2 y-i '-* where Λ is an arbitrary constant. Note that

/"' 1 1 7-1 J\ -fit-x

for all x lying in the interval (—1,1).

Example 6. Solve

*/W = (P) [ —A + g(x) J-\t- x

where k is an arbitrary constant, and f(x), g(x) are continuous real-valued functions defined on the interval (— 1,1). An integrable singularity for f(x) at x = ± 1 is also permitted.

Solution. Let

F(z) = — / A 2m 7_, / - z

Then

F+(x) - F-U) = /(*)

F+(x) + F-(JC) = —XP) í -í^-dt m 7_, t - x

Therefore the given integral equation reduces to

k[F+(x) - F-(x)] = m[F+(x) + F-(x)] + g(x)

F+W = -, :F-(x) + k — iri k — m We first find a nonzero function L(z), satisfying

L+(x) _ k + iri L~(x) k — iri

such that L(z) does not vanish on the complex z-plane. A suitable L(z) is

L(z) = —e®» z — 1

where

Now, if

then

_ 1 _ Γ _dt_ (k + m\ 2iri y_i f — z \k — mj

k + m = Vk2 + iP-e*, 0<β<π

k- m = y/k2 + -n2e~ßi

1 ¿ + τπ _ 2ßt _ p 2iri k — m 2m π

That is,

l / £ + m \ 27ri \& — 7ri /

x — I

< l

• .Jita*±g+ ■(f,)/i|"i»^o/c-">],; ά">/_ * — l I 2 k — m 2m ./-1 / — *

1 / * + TO ^

JC — 1 V & — 7ri

1 /* + 7Π (1 - jc)i-0/")(l + χ)β/τ \ k- m

where

2m \x+ \) \ \ +xj

Similarly

L_(x) = — — e°'M

x - 1

- 1 \k-mj V - i 2 i r i ( f -x )

THE HILBERT TRANSFORM ON THE REAL LINE 75

or

L-(x)

Clearly an appropriate L(z) has been found. Now the given equation can be written

1 x —

(1

l /*~ i n ' i -w 1 V k + m

1 - * ) 1 - -(0/ir)(l + X)Ph Jk~ <\lk +

■ni

m

L_(x) F+(jr) F_(jr) _ g(x)

U(x)

L+(x) L-(x) L+(*)(*-ir i )

Under the assumption that ^ = 0(1) as z —> <», the most general solution is

8(0 dt ) 2 m / . , ( f - . + *, z)L+(r)(* - m)

where ¿i is an arbitrary constant. We make the simplification

/(*) = F+(x) - F_(jt) = *,[L+(x) - L_(x)]

- Z._(jt)

1 *(*) + 2L+(x)(k — m) 2m

1 g(x) , 1 2L+(J:)(Á: — m) 2τπ

1 2

«w*

= *, [£.+(JC) - L-W] + - [L+(x) + L-(x)]

(t - x)L+(0(k - m)\ 1 g(t)dt ! (i - x)L+(t)(k - m)_

g(x) L+(x)(k - m)

+ [L+M-L.M}±u>)fl ff * ' > * 1 J2TTI y_, (f - x)L+(t)(k

L AM - m)

kg(x) 1 g"W f \ g{f)e-^dt *2 + 7T2 (* 2 4-7T 2 )U- l )^ V - l f - X

+ 27ri*i »ΜΊ

- 1 \ / * 2 + ΤΓ2

2.5. THE HILBERT TRANSFORM ON THE REAL LINE

Let / be a function defined on the real line, then its Hilbert transform (Hf)(x) is defined by


(Hf)(x) = -(/») / 1^-dt ■π 7_οο x - t

-^1(Γ\Γ\Μ-Λ (2.32)

provided that the limit exists. Our definition of the Hubert transform differs from that of Titchmarsch [99] in sign only, but this does not effect our basic results. Titchmarsh has discussed the properties of this Hubert transform in great length and has proved, among many others, the following theorems (the statement of these results in Tricomi [103] is especially clear).

Theorem 1. If the function f(x) G LP(U), p > 1, then the formula (2.32) defines almost everywhere a function g(x) = (Hf)(x) that also belongs to Lp and whose Hubert transform coincides almost everywhere with —f(x). In other words, we have the inversion formula for the Hubert transformation H in the space LP(R) as follows:

H2f = - / a.e.

The following relation between / and g hold

(Hf)(x) = g{x) a.e.l (Hg)(x)=-f(x) a.e.f K¿^}

Relations (2.33) are also called reciprocity relations between / and g. We now elaborate the concept of an analytic function in a region being regular.

Consider the function

/(z) = r z — i

The function /(z) is analytic everywhere on the complex z-plane except at z = i, where it is not defined. But we can prescribe the value 2/ to / at z = /' to make / analytic. Since the only singularity of / in the complex plane is a removable singularity the function, /(z) is said to be regular in the complex plane. From now on we can say that /(z) is regular in a region Ω if the only singularities of /(z) in fl (if any) are the removable singularities.

Theorem 2. Generalized Parseval's Identity. Let the functions /] (x) and fz(x) be-long to the classes LPl and LPl, respectively, p\,pz > 1. If

l + l - i P\ Pi

we have

/ Mx)Mx)dx= f [(Hf,)(x)][(Hf2)(x)]dx (2.34) J—CO J~ 00

Let

/,(*) = / ( * ) e L ' ( R )

and let

Mx) = (Hg)(x)

where g E. LP(U). Then we can derive from (2.34) and Theorem 1 the following formula:

/ f(x)(Hg)(x)dx = i [{Hf)(x)](H(Hg))(x)dx J— 00 J — 00

= f (-Hf)(x)g{x)dx (2.35) 7-00

In duality notation we can put this result into the form

<f.Hg) = (-Hf,g)

or (2.36)

W,g) = (J,-Hg)

Relations (2.35) and (2.36) are also called Parseval's identities.

Theorem 3. Let F(z) be an analytic function regular in the region Imz > 0 such that

\F(x + iy)\p dx < k (p > 1) /

where k is a constant independent of y. Then as y —> 0+, F(x + iy) converges for almost all x to a limit function u(x) + iv(x), which we denote by F(x + iO). The functions u(x) and v(x) are both Lp functions satisfying the reciprocity relations

vM = i(P) Γ ^-dt 7Γ J_x X - t

u{x)=-irmdt ΤΓ 7-00 x - t

(2.37)

Let us now apply this theorem to a function F(z) = - ^ . Clearly F(z) has no singularity in the region Im z > 0. It will be treated as an analytic function that is regular in the region Im z > 0.

F(x + iy) = x + i(y + 1)

1 [ \F(x + iy)\" dx = I ! y^dx J-°° J-" \x2 + (y +

Putx = (y+ l)tan0, - f < Θ < f:

fV2 , v + i)sec20 / \F(x + iy)\"dx= / ¿i—

y-» 7-W2 (y +

■L

/2(y+l)PsecPe ■n

cosp~2ede

άθ

Therefore the function F(z) = —. satisfies the requirement for the validity of Theo-rem 3,

F(z) = l

z + i 1 x — i lim F(z) = —— = = «(*) + iv(x)

>—ο+ x + / jt2 + 1

«w = x

v(x) =

x2+ 1 - 1

x2 + 1

Using the method of contour integration, it is easy to show that

- 1 x2+ 1 -m/.i?T7rV

which proves that v(x) = (Hu)(x). To these results of Titchmarsch [99] we add the following theorem:

Theorem 4. Let /(z) be analytic in the region Im z = y ^ 0 such that

/(z) = o(l), \z\ — °° uniformly V 9 £ [0, ττ]

Then

/(*) = i(Hf)(x)

that is,

W(JC) = ( / /H)(JC) (2.38)

u(x) = -(Hv)(x) V * £ l R

where / ( * ) = U(JC) + Í'W(JC).

Proof. By Cauchy's theorem it can be shown that

m = Λ / —dt (2.39) 2m J-«, t — z

This result is obtained by performing the integration of ^ ^ over the contour consisting of the real line segment —R < JC < /? and the upper semicircle with center at 0 and radius R and finally letting R —* oo. Denoting the semicircular arc by CR, we can see that

Jc. t - z >cR

For R sufficiently large and e > 0, we get

\JCR t - z I J R - \z\

lim/e_o. / dt\ s e \Jc„t-z |

Since € is arbitrary small number > 0,

no l i m / ^ -dt = 0 2

Result (2.39) follows. Letting z -» x in (2.39), we get

2m J_a t - x 2

or

m y.o, ί - x fix) = /(ff/)W

After equating the real and imaginary parts, we see the Hilbert receprocity relation (2.38).

To illustrate an application of Theorem 4, let us take

z + /' x + i(y + 1)

= x - i{y + 1) x2 + (y + l)2

So

fix + ¿0) = f{x) = X~l

uix) =

vix) =

x2+ 1 X

x2 + 1 - 1

x2 + 1

In this case it is easy to show by using the technique of contour integration that

vix) = (//«)(*) D

Tricomi [103] proves the next theorem by using the results proved by Titchmarsh [99] in Theorems 1 through 3.

Theorems. Let the functions <ρι(χ) and <P2(*) belong to classes Lp'[-l, 1], and LP2[-1,1], respectively. Then, if ^ + ^ < 1, we have

(//)[φ,(//<Ρ2) + <P2(//<P.)] = (W<Pi)(//<P2) - Ψ\Ψι

almost everywhere. Here (//)<p stands for the finite Hubert transform of φ:

iH)i<p)ix) = -iP) f -^-dt. ■π J-iX-t

We will now demonstrate the method used by Tricomi to find an inversion formula for the finite Hubert transform.

The finite Hubert transformation does not possess the uniqueness property. For ( " ) ( T = ? ) = o;

(//) - r = = = -(/>) / * }-—dt \\/\ -t2) * J-\ x~t

Putf = j^ ,y>0:

m[^[—^) TTJO il - x) - il + x)y2

_ 2 Γ dy + x)Jo

- 1 < J C < 1

TTU+X)J0 y2-(^x)

y/TTxy - y/l - x V ^

In v/l + *>> + \/\ -x

= 0 o

When U| > 1, } ^ < 0, we have

(//) VT TT(1

2 Γ c dy

x + 1 ir(l + x) V x - 1

1

tan

!)/(* + 1)

y sj(x - \)/{x + 1)

V i

1 - t 2 dt ' ir J-i yj\ - ,2 t

.-Lm ['-¿LJL * J-\ \f\-t2t~

' " « Í r * J-i Vi

w 7-1 V l - i 2

W f T J-\ ΛΤ

-.dt

We now can solve the aerofoil equation

f{x) = ~{P) [ ^-dt = (//)(<p) (2.40)

We want to find the solutions of this equation in the space Lp, 1 1, and that the function X— in the process of finding the solution must also belong to the

space LP, which is possible when p < 2. We take ψ](χ) - φ(χ), φι(χ) = v l - x2, ψι(χ) e LP2[-1,1 ], for any p2 > 1. By Theorem 5,

(//)[ίφ(ί) + V7! - f 2 / ( 0 ] = (//)[i<P,(f) + 92/(0]

= (//)[φ1(//)φ2 + <ρ2(//)φ1]

= [(//)φ,(//)(%)-<Ρι%]

= [*/(*) - Vl - χ2 φ(χ)] (2.41)

We obtain

(H)[t<p(t)] = - ( / > ) / ' X+tX<p(t)dt

= - - / <p{t)dt + xf{x) τ J-\

= -C+ xf(x)

Therefore from (2.41),

C + xf(x) + (H)[Vl-t2f(t)] = xf(x) - V\ - x2 <p(x)

Vi-t2f(o Φ) v/T -M i \ / l - x2 (x - t)

-dt (2.42)

To verify that (2.42) is a solution of (2.41) is a difficult task. For this we refer the readers to Tricomi [103] on the integral equation. By virtue of the fact that (//) , ' = 0, C in (2.42) can be taken to be an arbitrary constant.

This solution was also obtained using the technique of solving the Hubert problem. In fact, by using the Hubert problem technique, we can obtain many more solutions. This is because in the Hubert problem technique our classes of functions are associ-ated with analytic functions and not necessarily with the 2-0 class of functions.

2,6. FINITE HILBERT TRANSFORM AS APPLIED TO AEROFOIL THEORIES

We give below a brief account as to how Hubert transform is applied to wing theory as discussed by Robinson and Laurmann [83].

Figure 2.4 shows a pair of thin parallel, unstaggered aerofoils of equal chord. For thin aerofoils whose thickness effects can be neglected, the horizontal and the vertical

z-Plane

Figure 2.4

FINITE HILBERT TRANSFORM AS APPLIED TO AEROFOIL THEORIES 83

induced velocities on the aerofoil surface become

«' = -V ( ^ tan - + Ύ^ α„ sin ηθ n=\

v' = U I —- + y^«B cosηθ

v' = v — U sin a

u' = u — Ucosa

See Robinson and Laurmann [83, p. 142]. In the two-dimensional case the vorticity vector is always normal to the z-plane.

The complex velocity potential due to an isolated vortex of strength σ is given by

ισ π(ζ) = — log(z - z0)

The corresponding velocity is

w(z) = ισ 2ττ z — ZQ

(2.43)

(2.44)

We will assume that (2.43) and (2.44) for the case of two aerofoils forming a bi-plane are also possible for the self-induced velocity due to the vorticity distribution of each foil. For the vorticity distribution on the upper aerofoil, we put

γ, = -2«ί = W <*o tan ®*Σ>-sin ηθχ ■π > 0, > 0

The suffix 1 stands for the result concerning the upper aerofoil. A similar expression is taken for the lower aerofoil with suffix 2. The induced velocity at a point z0 of the upper aerofoil is found by integration of (2.44) over the vorticity distribution on the two aerofoils

He/2)+(¡h/2) ie/2)-(ih/2) w i ( z o ) = ^ / - ^ ΐ Λ ΐ + ^ /

¿TTl J-(e/2)+(ih/2) Z- ¿0 27Π J-(e/2)-*

r2(z) \dz\ (ιΛ/2) z z0

where h is the distance apart of the aerofoil and e is their length. Separating the real and imaginary parts in the foregoing result, we get the compo-

nents of the total velocity at the upper wing

1 i'n "i(zo) = Ucosa + —— I

2-Π- J-e/2

+ ± r" 2TT y_i /2

y\(x)(y - yo) n Uo - x)2 + (yo - y)2

■yi(-«)(y - y 0 )

dx

(x0 - χγ + (y0 - y)2 dx

and

<,Λ-η*\η + l f ΎΑΧ)(Χ0-Χ) J υ,(ζ0) = í /s ina + — / -r-— -?dx 2-n- J-e/2 (xo - xr + (yo - yr

j" + — f/2 y2{x)ix° ~x)

2TT 7_ t / 2 (xo - x)2 + (3Ό - y)2 Χ

or

w,(z) = (/cosa - — / - j - — - d x - ——-

and

ui(z0) = i /s ina + — / — 0 ι ,_ ~^άχ

J—t 2·"· J-e/2 (χο - x)2 + (yo - y)2

2w y-e/2 *o - x

Put

, = T The last integral in the last equation becomes

e/2 Λ , . ί ν \ i /■'

π y-f/2 xo ~ x π J-

ydte/2) at /2x0 - x 7Γ y_, 2x0/e - t

which is the finite Hubert transform of 7i(f). By working out the total force required to uplift the plane, we obtain the fi-

nite Hubert transform. Robinson and Laurmann [83, p. 187] have shown that the z component of the velocity induced by the bailing vortices at the point (0, y, z) in a three-dimensional aerofoil is

1 fSo

w(0,y,z)^-7-(P) ¡ -l^-dyi y i - y

It is assumed that the wing extends from y = — so to y = so·

2.7. THE RIEMANN-HILBERT PROBLEM APPLIED TO CRACK PROBLEMS

Inglis [92] in 1913 studied the distribution of stress in the neighborhood of an elliptic crack with semi-major axis c and semi-minor axis b contained in an infinite thin plate. Griffith considered the case where b = 0 when the elliptic crack degenerates into a straight line of length 2c. A straight-line crack can be considered as a degenerate case of an elliptic crack when b the semi-minor axis is zero. For this reason a straight-line

REDUCTION OF A GRIFFITH CRACK PROBLEM TO THE HILBERT PROBLEM 85

crack occupying the segment

y = 0, — c ^ x ^ c

is called a Griffith crack. Sneddon and Lowengrub [92] in their discussion on the crack problem in the classical theory of elasticity give a derivation by G. B. Kolosov [1909] to solve the two-dimensional equation arising in the field of two-dimensional elasticity and show how such equations were solved by Kolosov by introducing complex analytic functions. They state that "the major development of the present century in the field of two-dimensional elasticity has been Muskhelishvili's work on the complex form of the two-dimensional equations due to G. B. Kolosov [1909]."

2.8. REDUCTION OF A GRIFFITH CRACK PROBLEM TO THE HILBERT PROBLEM

Sneddon and Lowengrub [92] give a very good account of how Muskhelishvili used his technique of solving a Hubert problem to solve the complex form of the two-dimensional equations due to Kolosov [1909] arising in connection with finding the stress distribution in the neighborhood of a Griffith crack or a series of Griffith cracks. For details see [92, pp. 39-40]. The stress around a Griffith crack is related with function φ(ζ), satisfying the condition

[φ'(Λ:)]+ + [φ\χ)Υ = -¡Ax). x&L (2.45)

φ+(χ) - φ~(χ) = 0 xGX-L (2.46)

φ(ζ) = 0(ζ~'), a s | z | - * ° °

where L is the union of all the Griffith cracks lying along the Jt-axis or it may also be a single Griffith crack along the jc-axis. It follows that <p(z) is holomorphic in the complement of L, and therefore φ(ζ) is also analytic there and is 0(z~2) as z —* °°. Therefore the form of φ(ζ) is obtained by solving for the Hubert problem (2.45).

Let us now restrict to the case of one crack — c ^ x ^ c. By the technique discussed in the previous sections of this chapter, it follows that the solution to (2.45) is

2tn(c2 - z2y'2 J-c t - z

We can see that φ'(ζ) = 0 (j¡ j as |z| —► °°. When we integrate (2.47) to get <p(z), we take as zero the constant of integration in view of the asymptotic order of φ(ζ) as z —► oo. The condition (2.46) is automatically met. That is, φ(ζ) is analytic outside the line segment

—c^x^c, y = 0


2.9. FURTHER APPLICATIONS OF THE HILBERT TRANSFORM

2.9.1. The Hubert Transform

In 1987 R. K. Donnelly [35] gave a solution to the differential equation (singular differential equation)

Ρ(Τ),Η)ιι = / (2.48)

in the space T>' Π // '(D) where T> = jt and H is the operator of one-dimensional Hilbert transform and P(x, y) is a polynomial of finite degree in x and y. The space H'(T>) and Ί)' are defined in Chapters 4 and 1, respectively. It turns out that the space Ί)' Π Η'(Ό) is closed with respect to operator T> of differentiation with respect to / and the operator H of the Hilbert transforms. Donnelly also gave a solution to the operator equation

P(O,H)U = /

where

/ e O'(W) n H'(TKUn))

and P(x, y) = P{x\ ,X2,...,x„,y)i&A polynomial of finite degree in n + 1 variables x\, x2,..., x„, y- Ό stands for the operator

(I-./- -f)-V>i.V2 D.) \dx dx2 dx„)

H is understood to be the «-dimensional operator of the Hilbert transformation.

2.9.2. The Hibert Transform and the Dispersion Relations

The term "dispersion relations" comes from the observation of Kronig and Kramers in the theory of the dispersion of light by gaseous atoms or molecules. They observed that a relation exists between the real and the imaginary parts of the complex index of refraction, which is called a dispersion relation. This dispersion relation is closely associated with the Hilbert transform discussed in this chapter. Dispersion relations have been used to connect the real and imaginary parts of the scattering amplitude through Hilbert transform in high-energy physics. A good account of the role played by the Hilbert transform in the dispersion relation is given by Wu and Omura [107, Ch. 7]. Readers interested in the applications of the Hilbert transform to dispersion relations may look into [107]. Another good reference is the book by J. E. Marsden, Basic Complex Analysis, 1972, pp. 411—433.

EXERCISES 87

EXERCISES

1. Solve the integral equation

a(x)f(x) = k(P) f ψ ^ + g(x) 7-1 t-x

forx in (—1,1) where λ is real and > 0 and where a(x), g(x) are prescribed real functions. You may assume that / and g are both continuous functions in the interval (-1,1) and that / ( / ) is allowed to have integrable, singularity in the neighborhood of t = ±1 . Hint: Start with

2m 7_, t

F(z) may have algebraic singularity of degree > — 1 at the point ± 1, in which case

(a(x) - km)F+(x) = {Φ) + km)F-(x) + g(x)

and so on. The answer takes the form

a(x)g(x) /(*) = a\x) + λ2ττ2

sJa2{x) + k2TTlS' '7-1 (t - x)y/a2{x) + λ 2 π 2

ke*x)

+ ke^ / · ' g(t)e-«*>dt

(1 - x)y/a2(x) + λ2ττ2

where k is an arbitrary constant and

1 i D , /"' dt . a(t) + 2m J_i t — * α(ί) —

km km

2. Prove that if/(f) satisfies //condition on the unit circle |f| = l,theng(fl) = f(eei) also satisfies H condition in the interval (-00,00). Note that g{6) is a periodic function with period 2π.


nfWÍL + nfm. J0 t - x J2 t -

dt — = x x

for x either in the interval (0,1) or (2,3).

4. (a) Prove that

r l ' dx \/ι -χ2(χ- y)

( p ) / -7==7^-X = 0 V v e ( - l , l ) .

(b) Define

(///)(*) = !(/») f P^-dt rr }-\t-x

= 0 * £ ( - ! , 1 )

V ^ e ( - l , l )

Prove that (H2f)(x) = —f(x). Hint: Use the Titchmarsh inversion formula on the real line.

(c) If (///)(JC) = g(x), prove that /(*) = -(Hg)(x) + -j£—, where c is an arbitrary constant, — 1 < x < 1.

(d) Generalize the class of functions for which this result is true. 5. Show that the general solution of

mL 1 +P(t- to) f(t)dt = g(t0) [t-t0

where C is a simple closed curve and Pit) is a given entire function of t, is

/(/) = —L(/») /" i ^ l - -1 / g(T)/>(T - t)dr ΤΓ2 Jc τ-t TT2 Jc

6. Derive the solution to Example 4 from that of Example 5. 7. Let C, C£, and Ci be the curves as discussed in Figure 2.1. Assume that /(f)

satisfies H condition on C. Prove that Fp(t0) exists. We define Fp(t0) as

.. .. 1 /" fiOdt ,. 1 y /(f) . hm hm —- / -^— = lim —-1 / -^-^-dt. Í-.OZ-»(O 2m Jc-c, t — 2 e—o 2m Jc-c, ' ~~ Ό

//in/: Write /(f) = /(f) - /(í0) + /(f0).

3 THE HILBERT TRANSFORM OF DISTRIBUTIONS IN V{P, 1 <p<co

3.1. INTRODUCTION

Laurent Schwartz [87] developed the theory of Fourier transform of tempered distri-butions by using a Parseval-type relation as follows: Let / be a tempered distribution / G 5'. Then the Fourier transform Jf as a tempered distribution is given by the relation

<J/, = </.J V<pGS (3.1)

where the expression JF<p in (3.1) is understood to be the classical Fourier transform of <p. It is a well-known fact that the classical Fourier transform operator J is a homeomorphism from S onto itself. As a consequence the operator J defined by (3.1) is a homeomorphism from S' onto itself with respect to its weak (or strong) topology. One can use relation (3.1) to calculate the Fourier transform of tempered distributions.

Since the classical inverse Fourier transform operator J is also a homeomorphism from S onto itself, Schwartz defines the inverse Fourier transform operator on S' by using an analogous relation

<J"7. = </.J~V> V < p e S (3.2)

One can easily check that operators J and J , as defined by (3.1) and (3.2), are inverse transforms of each other.

The main objective of this chapter is to develop an analogous theory for the Hubert transform of distributions in D^, 1 < p < =». Shown will be many applications of this theory.

89

90 THE HILBERT TRANSFORM OF DISTRIBUTIONS IN 2 ^ , , 1 

In problems of physics we sometimes need to find harmonic functions U(x,y) in the region y > 0 whose limit as v —► 0+ does not exist in the pointwise sense but does exist in the distributional sense. The theory of Hubert transform of distributions that we are going to develop will provide an answer to the existence and the uniqueness of the solution to such problems.

The Hubert transform of distributions in various subspaces of T>' were investigated by a number of authors such as Schwartz [87], Beltrami and Wohlers [5,6], Gelfand and Shilov [43], Lauwerier [58], Mitrovic [61, 62, 63], Newcomb [68], Orton [72, 73], and Horvath [50]. Notable among the literature are the techniques followed by Orton [72, 73] and Schwartz [87]. Methods followed by Orton are dependent upon the analytic representation of distributions. This is not quite constructive for distributions not having compact support, and as such, the methods used by her cannot be applied with sufficient ease to applied problems dealing with computation of Hubert transforms of distributions having noncompact supports.

I now give a brief summary of the methods of Orton. She uses the fact that every / £ T>' has an analytic representation. In other words, for f G Ί)' there exists a function F{z) defined and analytic on the complement of the support of / such that

lim f [F(x + iy) - F(x - iy)]<pdx = (f(x),φ(χ)) V φ Ε Ώ (3.3)

This deep result was proved by Tillman [98]. Orton then defines the Hubert transform Hf{x) of a Schwartz distribution / G T>', relative to its analytic representation F(z), as the distributional limit Hf(x) defined by

Hf(x) = lim i[F(x + iy) + F(x - iy)] (3.4) y—»0

She easily justifies the nomenclature considering a distribution / of compact support; for in that case the analytic representation F(z) of / will be given by

and in this case the Hubert transform Hf(x) of / will be given by

Hf(x) = lim i[F(x + iy) + F(x - iy)]

= l i m — - ( f ( x ) , X~{

y^o m\ ' (x — t)2 + y2

= liin —f{x) * , , >->o τπ xl + yz

= >^*ρ·ν·0) ( 3 · 5 )

CLASSICAL HILBERT TRANSFORM 91

As the above demonstration indicates, Orton's method is not very useful in solving some applied problems involving the Hubert transform of distributions. Be it as it may, the theory developed by her is of great theoretical significance.

Laurent Schwartz [87] defines the Hubert transform of a distribution / G "D'y, as — (p.v. j ) * / where * stands for operation of convolution. His method involves very complex mathematical theories and cannot be used with sufficient ease to applied problems. Nevertheless his definition is of great theoretical significance.

The work done by Dragisa Mitrovic [61,62,63] in this direction is more oriented toward analytic representation of distributions, distributional representations of an-alytic functions, and distributional boundary-value problems, for example, and not toward the Hubert transform of distributions as such.

In this chapter I will develop a theory of the Hubert transform for the Schwartz distribution space Ό^, 1 < p < ° ° , that will be fairly simple and easily accessible to applied scientists. I will also develop the calculus of the Hubert transform on DL, 1 < p < oo( and demonstrate its application to solve some singular equations and integro-differential equations. Applications of this technique to solve some related boundary value problems will be also shown. I will use the theory to solve problems on analytic representation of distributions, distributional representation of analytic functions, and distributional Hubert problem. Related theoretical problems will also be discussed.

I will be using notation and terminology commonly used in analysis. The letter N will be used to indicate the set of nonnegative integers and the pairing between a testing function space and its dual is denoted by (/, φ). If / is a distribution then the notation f(t) is used to indicate that the testing functions on which / operates have t as their variable, but whenever no confusion arises we will tend to drop the argument of a distribution. The space of C°° functions defined on M. and having compact supports will be denoted by Ί) and not by T>(U.), but the space of C°° functions defined on W, n > 1, and having compact supports will be denoted by THU"). Throughout this book p is a real number greater than one, unless otherwise stated.

3.2. CLASSICAL HILBERT TRANSFORM

The Hubert transform (///)(*) at the p o i n t ^ G R o f a function/(/)defined a.e. on R is given by the following integral limit:

(///)« = lim - f Ά- dt (3.6) ί - .0+ 7Γ J X - t

U-»l>«

provided that the limit in (3.6) exists. The expression (Hf)(x) as defined by (3.6) is sometimes denoted by

— p.v. / dt IT J_„ X - t

92 THE HILBERT TRANSFORM OF DISTRIBUTIONS IN D^,, 1 < p < »

or by

1 Γ f(t\ dt 7Γ i - » X - t

The symbols P and p.v. are put to indicate that the above integrals are taken in Cauchy principal value sense. Some authors omit the factor Ι/π in the definition (3.6) and replace it by a " - " or a " + " sign. I will stick to the notation used in the definition (3.6) throughout this book.

Let us now consider briefly some of the important properties of the classical Hubert transform operator H defined by (3.6), for which we will briefly recall some results of Chapter 2.

1. If / E if, K p < o o , then (Hf)(x) exists a.e. on U, and it belongs to Lp. As a matter of fact the limit in (3.6) exists in if sense too [99]. There exists Cp > 0 independent of/ such that

\\Hf\\p s Cpll/Hp

2. If /(f) and g(t) belong to if, and if' respectively, where 1 < p < » and P' = Ρ/ÍP- l),then

/ (Hf)(x)g(x)dx = - / f(x)(Hg)(x)dx (3.7) J— (X J— 00

In duality notation, relation (3.7) can be written

W,g) = (f,-Hg) (3.8)

Note that the existence of the integrals in (3.7) follows from property 1. 3. Let H be the operator defined by (3.6). Then

(H2/)U) = -f(x), a.e. (3.9)

and in L? sense as well V / G L ' , l<p<°°. Relation (3.9) can be expressed by saying that on if, 1< p < °°,

H2 = - / (3.10)

4. If / e if, Kp < oo, then u(x, y), v(x, y) defined by

u(x,y) = - ., , 2dt ir J-oo (' - x)2 + y2

v(x,y) = - / . , , 2dt ir /_„ (r - x)2 + y2

Both functions exist for y Φ 0 and belong to if when treated as functions of x for each fixed y Φ 0. Functions u and v are also conjugate harmonic functions

SCHWARTZ TESTING FUNCTION SPACE, V>L„ 1 < p < » 93

in the region y Φ 0. Furthermore

lim W(JC, ? ) = / ( * ) (3.11)

lim v(x,y) = -Hf{x) (3.12)

where the limits hold in if sense and also a.e. [88]. 5. If / e L2, then the integral lim¿v_<» J_N f(t)e'xl dt exists. That is, there exists

a function in L2 to which the limit as lim/v_<» f_N f(t)e"" dt converges in L2

sense.

We denote the limit of the integral in property 5 by (Tf)(x) and call it the Fourier transform of / . Thus

N

( J / ) W = l.i.m. / f(t)eix'dt (3.13) ¿V-.00 J

-N

Let J" be the operator defined by (3.13) and H be the operator defined by (3.6), then [75]

(JHf)(x) = i sgn(x)(J/)(jc) V / e L2 (3.14)

In this chapter our theory of distributional Hubert transform will be centred upon the Parseval-type relation (3.8).

3,3. SCHWARTZ TESTING FUNCTION SPACE, Τ>ύ, Κρ< »

A function <p(t) defined on U belongs to the space T>^ iff

1. φ(ΐ) £ C°°. 2. <p(4)(') G Lp V k ε P*J; N is the set of nonnegative integers.

Note that T> and S are both subspaces of DLP , 1< p < «.

3.3.1. The Topology on the Space DLf

The topology on the space T)^ is generated by the separating collection of seminorms {%}teNl [1,108, 110], where

yk(<P) = (J \<Pw(t)\"dt\

Therefore sequence {<pM}^=1 converges to an element φ in DLP as μ —> °° iff

7(*)(<Ρμ -<ρ)—»0 ββμ—>o°V/te(^J

A sequence {<ρμ}^=, is said to be a Cauchy sequence in DLP iff for all it e M,

■v*(<Pm - <pn) -* 0 a s m , « - » «

94 THE HILBERT TRANSFORM OF DISTRIBUTIONS IN &L„ l

independently of each other. It is well known that DL?, 1 < p < °°, is a locally convex, Hausdorff and sequentially complete topological vector space [87,67]. It is also known that if φ £ V^, 1 < p < °°, then Ηπΐ|χ|_„ <p(k\x) = 0 for each i t G N [87,42, 67].

If {<ρμ}" , is a sequence converging to zero in D^, then it is well known that the sequence {ψμ\θ}^.= ι converges to zero uniformly on IR for each k £ N [87,42,67]. Therefore, for each € > 0, 3n £ M independent of / such that

\<p«\t)\<e V M > n (3.15)

Note that n may depend on k and φ. In the sequel for 1 < p < « we will write p' = p/ip — 1) and denote by Dtp the

dual of the space D „,; that is, D[, = (25 , .) ' and so (Ify)' = D'u, It can now be seen quite easily, in view of the above-mentioned properties, that

a distribution with compact support is an element of Dtp. We have D C D¿ and convergence of any sequence in D implies its convergence in D^p. Therefore the restriction of / £ Dtp to D is in D'. In view of the fact that D is dense in DJP [87, 67] it follows that there is one to one correspondence between / £ "Dip and its restriction to D. Consequently

D'DD¡P, K p < o o (3.16)

We now prove the following:

Theorem 1. Let H be the operator defined by (3.6). Then, for 1 -* 7Γ e—0+ J X - t

| / - * I > Í

= - 1 lim / «*±ϋ>Λ π «-»0+ J t

: - I / «*±Ξ>Δ - I u« / * ± - Λ (3.19) 77 J t 77 (-.0+ y t

\t\>N Na|r|>e

. - I / « * ± ^ , - - L „ m / * ' + «>-««>* (3.20) TT J t 7Γ Í-.0+ J f

*)

W>N

. . I ί«±Ξ>Λ-1 [ π J t π J

N==|;|>£

ψ(χ, f) A (3.21)

\Λ>Ν N*.\t\

SCHWARTZ TESTING FUNCTION SPACE, J>L„ 1 < p < »

where

( φ{ί + χ)- φ(χ) t

<p'(x),

¡Φ0

t = 0

95

(3.22)

Clearly ψ(χ, t) defined by (3.22) belongs to C°°(R2), so (3.21) is justified. It is easy to see by Holder's inequality that

/»W = I ^L±Adt ■\>N

w I N~U"

\p ( 1 -piy/p' (3.23)

Therefore the sequence in ¡N(X) converges uniformly on U as N —> °°. Hence invoking standard theorems on the uniform convergence of of integrals [48], we get from (3.21),

(Ηφ) ψ(χ,ι)ώ

Φ(χ,ί)ώ \,\>N

77

¿ /«^±*)Λ_ι f ΊΤ J t π J

\t\>N Nz\t\

= - ! / í£±i>A - 1 1¡m f π J t 7Γ Í—o+ J

7 / 7 Γ ί - 0 + J t \t\>N N>\t\>t

= - i / ^ ± ^ d í - i l i m / *£±ϋ>Λ TT J t 7Γ í—0+ 7 /

|(|>Λί /Va|<|>e

= -ÍHm (&Ϊ*Λ 7Γ Í—O+ y /

Mx

= ilim [™* π €—o* J x - t

= (Ηφ')(χ)

Using a similar technique, we can show by induction that

(H<p)(k\x) = (H<pik))(x) V k £ N

By the properties of H quoted in Section 3.1, we get from (3.24)

||t**ni,*c,||«H, v * e f *

(3.24)

(3.25)

96 THE HILBERT TRANSFORM OF DISTRIBUTIONS IN D^,, 1 ^, \<p<<*>. Linearity of H is trivial to prove. H is a linear mapping from T>^ onto itself.

To prove that H is one to one, assume that for φ G T>^, (Ηφ)(χ) = 0. Operating on both sides of the above equation by H, we get Η2φ = 0. Thus — φ = 0; that is, ψ = 0. So Ηψ = 0 => φ = 0. Now we show that H is onto.

Let φ G 2\?. Then (-Ηφ) G T>t. But Η(-Ηφ) = - Η 2 φ = φ. Therefore for every φ E DLP there exists — Ηφ G DLP that is mapped by H to φ. Consequently H is also onto.

Clearly H is a one to one and onto mapping from Ό^ onto itself. Therefore H is defined on DLc Since — H2<p = φ for all φ Ε T)¿, it follows that

H"' = - H (3.26)

The continuity of H and H follow from (3.25) and (3.26), respectively. D

3.4. THE HILBERT TRANSFORM OF DISTRIBUTIONS IN DJ,, K p < »

In analogy to Parseval's relation (3.7) for the classical Hubert transform operator H, we define the Hubert transform Hf of / E T)L by the relation

(Hf, φ) = {f, -Ηφ) V φ E T> ,, p' = -^— (3.27) L p — 1

Clearly the Hubert transform Hf of / E T>L is a functional that assigns the same number to ψ as / assigns to — Ηφ, where Ηφ is the classical Hubert transform of ψ as defined by (3.6). The linearity of H on D', is trivial and the continuity follows by the result proved in Theorem 1.

3.4.1. Regular Distribution in ü j ,

Let / E Lp, \<p <oo. We can define

< / » = f f(x)<p(x)dx ν φ Ε ϋ , , (3.28)

Linearity and continuity of/ defined by (3.28) can be easily proved. Let us now find its distributional Hubert transform Hf. By definition,

<Hf,<p) = (f.-H<p)

= - f_ HxW<p)(x)dx

= / (Hf)(x)(p(x)dx (Parseval's relation)

Hence

THE INVERSION THEOREM 97

3.5. THE INVERSION THEOREM

Theorem 2. Let / £ V!¿, K p < ° ° . Then

- H 2 / = / V / E D ¿ (3.29)

The generalized Hubert transform H defined by (3.27) is an isomorphism from Ό^ onto itself, and H = — H is valid on Ό'^.

Proof.

< H 2 / » = <H(///),cp>

= (Hf,-H<p) V<p&O¿,p' = p/(p-\)

= </.(-Η)(-Η)φ> = {~f< ψ) V φ £ D „i (by inversion formula)

Therefore H 2 / = - / :

H2 = - I (3.30)

By definition, if / £ £>(,, then Hf £ B[,. Now, if/// = 0, then H2/ = 0 and by (3.30), / = 0. Put differently, Hf = 0 =► / = 0, so H is one to one.

The fact that H is onto follows by (3.30). Indeed for every / £ T)'^ there exists (-Hf) £ O¡p such that H ( - / / / ) = / .

The linearity of H is trivial. Therefore H is a linear isomorphism from DJ, onto itself. H exists and in view of (3.29)

H"' = - H (3.31)

Corollary. For 1 <p < °° the generalized Hubert transform H defined by (3.27) is a homeomorphism from T)L, onto itself with respect to the weak as well as the strong topology on T)L.

Proof. In view of Theorem 2, we need only show that H is continuous. We will prove the continuity of H with respect to the weak topology on T)L. The corresponding results for the strong topology on OL will be discussed in Chapter 5 where H is defined as a distributional convolution with £ p.v.£. In fact it will be proved in Chapter 5 that the latter convolution coincides with the Hubert transform considered here. So the proof given in Chapter 5, for the continuity of H in the strong topology of J)ipy will be applicable here too.

Assume that /„ —♦ 0 in T>L. Then

{Hfv, Ψ) = <Λ, -Ηφ) V φ £ D , ,

—► 0 a s v —» oo

That is, ///„ —» 0 weakly as v —» ». This completes the proof of Theorem 2. D

98 THE HILBERT TRANSFORM OF DISTRIBUTIONS IN T/L„ 1 V <p e D , , (3.32)

Then

(HD*/) = D*///( V it £ N (3.33)

Proof. We prove (3.33) for the case it = 1, and the general case can then be proved by induction.

<HD/, φ) = (Of -Ηφ) V ψ e DL, , , p ' = p / (p - 1 ) , l</><°°

= (/, -D(-//<p))

= </,ϋ//φ>

= </,HD (by [24])

= ((-D)(-// /) ,<p)

= (D(///),<p) V ^ £ I \ ,

Thus we have proved that / / D / = D/ / / . D

3.5.1. Some Examples and Applications

Example 1. Show that H8 = i ρ.ν.γ.

Solution.

(Ηδ,φ) = <δ, -Η«ρ> V φ e D ^ , ρ' = ρ/(ρ - 1), 1<ρ<°ο

- / f t - I l i m / *>-Δ\ λ ι*-<ι>£ '

= - i l i m / * ! > * π €->ο* J —t

Ι»Ι>*

-ρ.ν.-,<ρ(0 ΊΤ I

Therefore

Η8 = - ρ.ν. i (3.34) 7Γ /

THE INVERSION THEOREM 99

Example 2. Find//(p.v. j ) .

Solution. Operating on both sides of (3.34) by H, we get

Η 2 δ - ί Η Η) — δττ = H I p.v.- I

Therefore

/ i \ ■ττδ H (p.v. j ) = -,

Example 3. For / e T>'^, 1< p < °°, find a solution to the operator equation

y(x) = Hy+ f(x) (3.35)

where H in (3.35) is the generalized Hubert transform operator. Show that this is the only solution to (3.35).

Solution. Operating on both sides of (3.35) by H, we get

Hy = H2y + Hf

y(x) - fix) =-y + Hf _ f{x) + Hf

y 2

Uniqueness: If yi and y2 are two solutions to (3.35), then it follows that H(y\ — y2) = 0Ί - n) o r "Cyi - yi) = ~(>Ί - yi). By addition we get H(yx - y2) = 0 => yi - y2 = 0.

Example 4. Solve the following integral equation if / G if, l < p < ° ° :

Hy = lim f 1^-dt + S(x) + p.v.- + -^— (3.36) Í-.0+ J t - x x x1 + 1

Solution. We can rewrite (3.36) as

Hy = -irfif + 8(x) + p.v. i + - J — (3.37) x xl + 1

Operating on both sides of (3.37) by H and using Examples 1 and 2, we get

H2y = - T T H 2 / + - p . v . - - πδ + IT X ■(=¿0

100 THE HILBERT TRANSFORM OF DISTRIBUTIONS IN Z ^ , , 1 < p < »

or

y = - i r / -Ip.v.I + w B - H ( ? l T )

Using the technique of contour integration or otherwise, we get

(3.38)

Therefore

- l i e - t y(t) = -rrf - - p.v.- + ττδ - - j — -

7Γ t f- + 1

Example 5. Solve in T)L, 1 < /? < °°, the operator equation

dy , + //8'(x) = 2xe~x (3.39)

a*

Solution. Using the properties of H, we can write (3.39) as

£,+«,.-!(,-.■) so

y + / /δ + e'"2 = *

The only constant distribution that belongs to T>'^ is the zero distribution and therefore k = 0. Hence

= - ( i p - v - i + e " O 3.6. APPROXIMATE HILBERT TRANSFORM OF DISTRIBUTIONS

For 1 L and for each fixed η > 0, let us define the numerical valued function F,, by the relation

^ ( / C ) . ^ ) ^ ( H , / ) « (3.40)

It is a simple exercise to show that for any fixed real x and η > 0, (x—t)/((x—t)2 + i)2) as a function of t belongs to T> „*, p' = p/(p — 1) > 1.

Therefore the function F,, as defined by (3.40) exists (i.e., is meaningful). Since the space Ό^ is closed with respect to differentiation, it follows that, for any η 6 Α

APPROXIMATE HILBERT TRANSFORM OF DISTRIBUTIONS

and η > 0,

101

dtk x - t

(x- t)2 + rj2 T>.. Vk

as a function of /. Now by using the technique used in [42] or the structure formula for / [47, p. 201], we can show that

1 dk x - t dxkF<n \ / 0 · π dxk (JC _ , ) 2 + η2

We now wish to derive a duality relation for F,, and show that

(3.41)

lim F„ = Hf in T>'.P

in the weak topology of D^. We now prove a lemma that will be useful in the sequel.

Lemma 1. Let / £ D'P, 1 < p < °°, and let Fv be the approximate Hilbert transform of / as defined by (3.40). Then for each kGN Ff(x) G Lp.

Proof. We will sketch the proof for k = 0. In other words, we will show that Fv G Lp, and the proof for the case k > 0 can be given in a similar fashion by induction.

In view of the structure formula for / [87, p. 201], there exists a nonnegative integer r and functions / G if such that

Ρ ' -Σ/>(-Ϊ) i - l

x - t (x - t)2 + rf

dt (3.42)

It is a fact proved by Titchmarsh that if / ( / ) £ L , then the expressions

x - t

and

/ J -a

f

MO

flit)

(x-t)2 + r?

1

dt

(t - x)1 + η 2

belong to if [99, p. 132]. Therefore, using the fact that

x - t

dt

(X - t)2 + T)2 1

2|η|

(η real Φ 0) in majorising derivatives in (3.42), it follows that each of the terms in the summation in (3.42) is in if. D

102 THE HILBERT TRANSFORM OF DISTRIBUTIONS IN CL,, 1 0 by (3.40). Then

lim F„ = Hf(x) η - - 0 +

in the weak topology of T>L.

Proof. In view of Lemma 1, we know that

Ov<p<*»-/e Fv<p(x) dx

exists for all φ G T> „> and that the regular distribution generated by F,, belongs to T>{p. Using the relation (3.42), we have

/

oo Γ - ΛΟΟ

*(/>H) x - t dt (3.43) (χ-ί)2 + η2

In view of Fubini's theorem the right-hand side expression in (3.43) reduces to

<p(x)dx Σ ;£'<<'> |/".(-0 [ΪΠ^ΤΙ? dt

.{x-tY + i? dx) dt

( - - : / : Ψ(Χ)(Χ - 0 dx

(x - 02 + η2

= (/(/), -dV)(f)> V «p G Ü , (3.44)

Since φ 6 D ; - , in view of Lemma 1 it follows that Ηη<ρ G O , / . This justifies the existence of the expression in (3.44). Now

φ{χ){χ - t) (X - i)2 + η2 dx

x - t

<? N

(x - i)2 + rf ¿χ

¿je = LjiX){-¿xJ [(X - O2 + η2.

/" X - í a>(i)(.x) z :r dx (by integration by parts)

(JC - t)2 + η2

APPROXIMATE HILBERT TRANSFORM OF DISTRIBUTIONS 103

for limîâ, φ(χ) = 0 whenever φ G T> ,<, 1 ,/ as η -► 0+ [70]. Hence letting η -► 0+ in (3.44), we get

lim+ (Ηη/, φ) = lim </, -Ηηφ> = (/, -Ηφ) = <Η/, φ> η—>0+ η—>0+

so limô* Η η / = Η / for all / £ D [ , . D

The following theorem can be proved in a way similar to that followed in proving Theorem 4.

Theorem 5. L e t / e D . ' , , \<p<°°. Then for each φ e Ώ.,,ρ' = p/(p - 1), we ^ L

have

/- "b+ \ \ v ' " 7Γ (i - JC)2 + y

that is,

h™ (< / ( ' ) . - ,. _ „;2 , „2 ). <PW ) = </. V φ e Dif

lim / / ( f ) , - y^-—2 )=f in ©[, (weakly) (3.45) y->0+ \ TT(t - xY + yl / L

3.6.1. Analytic Representation

Let / €= D^, 1 < p < oo, and let F(z) be the complex-valued function of z defined for Im z # 0 by the relation

F(z) = ~(f(t),-^-) (3.46) 2-7Π \ t — z I

Using the structure formula for / and a technique of Lemma 1, we can show that

Fw(z) = - L / / ( f ) > *! \ Imz * 0, * G N Ζ7Π \ (t - Z)"+1 I

104 THE HILBERT TRANSFORM OF DISTRIBUTIONS IN O^,, 1 < p < «

We see that F(z) is analytic in the region Im z Ψ 0. Using Theorem 4, we can show that

lim+(F(jf + ie) - F(x - ie), <p(x)) = </, φ) V φ G Ό ,, ρ' = -?— €—►() *-■ p 1

Therefore F(z) defined by (3.46) is the analytic representation for/ G D',, K p < ° ° .

3.6.2. Distributional Representation of Analytic Functions

Using the structure formula for / ε T>L,p' = p/(p — 1), 1 0 satisfies the following relations:

1. For any δ > 0,

sup \F(x + iy)\ = As < » (3.47) — oo<jr<ao y==6>0

and

^ + '»I = ° (y^) ■ ^ - · ; ~ *jr (348)

Let us now reverse the problem. Let F(z) be analytic on the open upper half-plane and satisfy the relations (3.47) and (3.48) such that F(x + ie) E L2, for any e > 0 and lime_0

+ f(' + '«) = f+(0 in £>[,, K p < » . Can we find/ E 23[P such that

F(z) = / / ( i ) , - ^ ) . Imz > 0 (3.49)

The answer to this question is affirmative. It is provided by the following theorem:

Theorem 6. Let F(z) be an analytic complex-valued function of the complex variable z = x + i y in the open upper half-plane (Im z > 0), satisfying

i. for fixed y > 0, p' = p/(p - 1), 1 <p<°° ,

F(x + iy) E Lf (3.50)

ii. lim^o* FU + iy) = f+U)in Dif (weakly),

sup |F(JC + iy)\ -> 0 as y -> °° (3.51)

sup \F(z)\ = A8 < oo (3.52) -oo<jr<oo y>S>0

APPROXIMATE HILBERT TRANSFORM OF DISTRIBUTIONS 105

Then

^) = (¿/+(0,-L-i ¿m t — z

V R e z > 0 (3.53)

Proof. Let us do the complex integration of the function

1 F(t + ie) 2~ϊή (ί - z)

with respect to the variable t along the semicircle of radius R, center at the origin lying on the upper half-plane with the diameter of the semicircle lying on the *-axis. Let A and B be the points at the left and right ends of the diameter on the *-axis. Choose points A' and B 'on the circumference of the semicircle lying in the second and the first quadrant, respectively, such that ΔΒΟΒ' = LAO A' = η > 0. By using Cauchy integration, we have

2m J t - z 2m J t - z -R ArcBB'

+ -L / FJL± 2m J t-

ΑκΒΆ'

1 I FJL±ieldt 2m J t — z

«) dt

AucA'A

= F(z + ie) l m z > 0

Taking/? > \z\,

_1 2 m

[ F{t +

i J t-" ) dt

ArcBB' 2m

Similarly

2m J t - z Arc A'A

2m

yRAf

(R - |r|)

yRAt

(R - \z\)

lim 2m J t-z AscB'A'

,. I 1 I (-Π- - 2η) /? - i i mk~^ p ii SUP F w

y — » 0 0

(3.54)

(3.55)

(3.56)

(3.57)

106 THE HILBERT TRANSFORM OF DISTRIBUTIONS IN 2 ^ , , 1 °° in (3.54), and using (3.55), (3.56), and (3.57) and then in turn letting η —► 0+, we get

1 JF(t + ie) (F(z + ie), lmz>0 2m J t-z dt \0 , lmz<0 ( 3 5 8 )

— oo

That is,

/ » F ( / + / e ) > _L\ = / ^ + »). ¡mz>0 \2τπ t-z I \0, Imz <0

Letting e -+ 0+ in (3.59), we get

/ l f+(A ! \ _ JF(z), lmz>0 , „ . V ^ 7 ( , ) ' Γ ^ / - \ 0 , lmz<0 ( 3 · ^

The uniqueness of F(z) is evident in view of the representation formula (3.60). □

We will now establish a further result pertaining to the case Im z Φ 0.

Theorem 7. Let F(z) be an analytic complex-valued function of the complex variable z = x + iy in the complex plane cut along the real line satisfying, for any fixed y Φ 0,

F(x + iy)Glf, Kp<cc (3.61)

lim F(z) = / ± (x ) inD!, (3.62) >->o± L

sup |F(z)| -► 0 as H -► °° (3.63) -00<JC<00

sup |F(z)| = As < co (3.64) -oo<jr<oo >>6>0

Then,

F(z)= ( ¿ ( / + ( 0 - /"( ' ) ) . ^ V lmz*0 (3.65)

Proof. As in Theorem 6 we prove that for Im z > 0,

^ιτΓ t-z I \0, l m z < 0

Similarly, performing the integration of the function

1 F(t-ie) 2m t — z

THE SOLUTION TO A DIRICHLET BOUNDARY-VALUE PROBLEM 107

in the semicircle in the lower half-plane and taking appropriate limits, we get

We arrive at (3.65) by combining (3.66) and (3.67). The uniqueness of F(z) follows in view of the representation formula as given by (3.65). D

Further results dealing with Hardy spaces and dealing with questions similar to those considered in Theorems 6 and 7 will be discussed in Chapter 6.

3.7. EXISTENCE AND UNIQUENESS OF THE SOLUTION TO A DIRICHLET BOUNDARY-VALUE PROBLEM

This section gives the solutions to some Dirichlet boundary-value problems related to the Hubert transform of Schwartz distributions. I first present some definitions, examples, and a theorem on these problems.

Definition. A harmonic function w(x,y) defined on the open upper half-plane (Im z > 0) is said to belong to the space J{p if and only if for fixed y > 0,

H>U,y)eLp, \<p<oo (3.68)

when treated as a function of x,

exists in the weak topology of T)L

lim w(x,y) (3.69) y - > 0 +

sup \w(x, y)\ = As < oo (3.70) — oo<jc<oo

sup |w(jc,y)|->0 as y -» oo (3.71) -oo<jc<oo

We now prove the following:

Theorem 8. Let u(x,y), v(x,y) be the conjugate harmonic functions belonging to the space 3{p as defined above. Assume that

limv_0+ u(x,y) = / ] y ° ' J \ weakly in Dp (3.72) hm^o* v(x,y) = g) L

then we have

Hf = ~g J / (3.73) Hg = f

108 THE HILBERT TRANSFORM OF DISTRIBUTIONS IN 0Lr, 1 < p < «

Moreover

and

«(*. y) = - (f(t), 1—j ) (3.74) 7Γ \ (/ - xy + y2 '

v{x, y)=- (fit), ' X ) (3.75) 7Γ \ (f - x)2 + y21

More clearly u and v are the only harmonic functions belonging to $ip satisfying (3.72) and (3.73). The convergence in (3.74) and (3.75) are interpreted in the weak topoplogy of "D^.

Proof. Since u(x,y), v(x,y) are conjugate harmonic functions in the upper half-plane, the function F(z) = u + iv is an analytic función of z = x + iy, and by Theorem 7,

«i(x. y) + iv(x, v) = / / ( / ) + ig(t), 1-. - L - \ (3.76)

1 (/ - JC) + iy 2τή (t - x)2 + y2 /(0 + i g ( 0 , ^ ; . _ r ¿ ^ ) i3·77)

Letting y —» 0+ in (3.77) and using Theorems 5 and 6, we get

f + ig=-j.H(f + ig)+l-(f + ig)

Therefore

( / + ig)i + H(J + ig) = 0

or equivalently

(Hf -g) + i(Hg + / ) = 0 (3.78)

Since / and g are limits of harmonic functions, they assign real values to the real-valued elements ofT>pi, 1^p, we get

[/ - * = ol + Hg = 0J Ψ. \ : ;> (3.79)

Since the functionals involved in (3.79) are linear, (3.79) also holds on complex-valued elements of T),p.

THE SOLUTION TO A DIRICHLET BOUNDARY-VALUE PROBLEM 109

Now equating the real and imaginary parts in (3.77), we get

u(x, y) = //(O, -Í- ,. Z , . ,) + (g(t), ' ' " * 2TT (i - x)2 + v2 / VV '' 2ir (/ - x)2 + y2

= \fit),2^(t-x)2 + y2)

+ ((Η/)(ί)·έα^Γ7) tby(79)]

= a ( / ( , ) ' ( i - ^ + y2 + H{( i -V+y2}) = ±(m. y ■ y

2TT \ J {t- x)2 +y2 (t- x)2 + y2

7Γ \ (/ - x)2 + y2 /

Similarly, by equating the imaginary parts in (3.77) and using (3.79), the Hubert reciprocity relations, we get

^ . v ) = i ( / ( 0 . ( f _ y + y 2 ) (3.81)

It is easy to verify that u and v as obtained in (3.80) and (3.81) are conjugate harmonic functions. An appeal to the structure formula for / G. D^ shows that there exist constants C and C such that

sup Iu(x,y)\ < C—ry as y—»oo *eR r / p

sup \v(x,y)\ ϊ£ C'-rj- asy—oo *eR yup

sup \u(x,y)\ = As <oo yS:S>0

sup \v(x,y)\ = Bs <oo — <n<X<<x>

Therefore W(JC, y) and I/(JC, y) as defined by (3.67) and (3.68) belong to the space 3ip. The uniqueness of the conjugate harmonic functions u and v satisfying the desired result follows in view of the representation formulas (3.67) and (3.68), respectively. D

Example 6. Solve the following Dirichlet boundary-value problems in the space T>L, (interpreting limits in T>L) 1< p < oo.

110 THE HILBERT TRANSFORM OF DISTRIBUTIONS IN X^,, 1 0, H £ J f

such that

lim u(x, y) = — p.v. f - ) in 2λ', >—0+ 7Γ \Jf/ L

Solution. Rewriting the problem, we get

V2w = 0, y > 0, M G 3Í

lim ίφτ,ν) = Ηδ y->0+

The obvious solution is

u(x,y) = ( M ( Í ) . - ; π (f - Λ:)2 + y2

3.8. THE HILBERT PROBLEM FOR DISTRIBUTIONS IN 2^,

3.8.1. Description of the Problem

Let us consider first a description of the classical Hilbert problem. Let g(x) be a function defined on the real line. We wish to find a holomorphic function G(z) analytic on the complex plane cut along the real line, such that

lim G(x + iy) + lim G(x + iy) = g(x) (3.82)

y—Ό* y—»0

Denoting the two limit functions in (3.82) by y+ and γ_, respectively, we have

y+ + γ. = g(x) (3.83) In many problems of mathematical physics slightly more difficult problems appear:

Given functions g(jt) and k(x) defined on the real line, we wish to find a function G(z) analytic on the complex plane cut along the real line such that

y+ + k(x)y- = y (3.84)

where y+ and y- are the limits of G(z) as described above. We explain formally how the problem in (3.84) can be reduced to a simpler form (3.83) by using the factorization technique. Essentially, the method consists in finding a function k(z) analytic on the complex plane cut along the real axis such that

log*(jc) = logAT+(jc) + log K- {x) (3.85)

where

K+(x) = lim k(x + iy)

THE HILBERT PROBLEM FOR DISTRIBUTIONS IN 2 ^ , 111

and

K~(x) = lim k(x + iy)

Note that the problem (3.85) can be solved by the same technique used in solving problem (3.83), and it also implies that

k(x) = K*(x)K'(x) (3.86)

(3.86) is called the factorization ofk{x). Using the factorization (3.86), we can now reduce the problem (3.84) to the form

y+ +K+(x)K-(x)y- = y Ύ+ + K'(x)y- = - } - (3.87)

KHx) "- K+(x)

Thus the problem (3.84) has formally been reduced to the form (3.83). Therefore our prime interest will be in solving problem (3.83), which we call the Hubert problem. Lauwener [58] considers the solution to the same problem, taking g(x) as some regular, or even a singular, distribution with compact support. We will find a function G(z) analytic in the complex plane cut along the real axis satisfying (3.83) where

y+ = lim G(x + iy) in the weak topology of T)¿ y—*0+

and

y~ = lim G(x + iy) in the weak topology of 2V >—o

Mitrovic [61, 62, 63] considers the Hubert problem in the space 0'a, a < 1, but he interprets limits y+ and y- in the weak topology of T>'.

We now rephrase the distributional Hubert problem.

3.8.2. The Hubert Problem in Z^,, Kp< co

Let g(x) G T)'^, 1 ', weakly (3.89) y—>0+ L

y- = lim G(x + iy) in £>', weakly (3.90) >—»o- L

112 THE HILBERT TRANSFORM OF DISTRIBUTIONS IN &L„ l 0 (3.91) 2τπ \ t — z /

Therefore

G(z) = r ^ /*W. „ ' " * 2 \ + T " (*('>< T. Ϊ Π Γ Τ " ) <3·92> 2τπ \ (ί — Λ:)-1 + y¿ / 2π \ (ί — χ)ζ + y1 /

Letting y —> 0+ in (3.92), we get in the sense of weak limits in Όίρ,

y+ = ~hHg + l8 (393)

Let us now define

Therefore

G(z) = - ~ (g(t), - i - > , Im z < 0 (3.94) 2m \ t — z

G(z) = -L (^-ϊΓ^τψ) ~ h (*'>-F4T7) (3-95) Letting y —► 0+ in (3.95), we get in the sense of weak limits in T>'^,

Ύ- = +γΜ8+1-8 (3.96)

Adding (3.93) and (3.96), we get

y+ + y- = g (3.97)

If, however, we relax the conditions in (3.89) and (3.90) and interpret the correspond-ing limits in T>', then there will be infinitely many solutions to the Hibert problem given by

G(z) = - L /g(i). γ^-ζ \ + F(z), Im z > 0 (3.98)

= - ~ (git), T^—)- Hz), Imz < 0 (3.99) 2m \ f — z I

where F(z) is an entire function. If in addition we impose the condition that F(z) is of finite degree at infinity, then

G(z) = ^ - . (g(t), -^—)+ P(z), Imz > 0 2m \ t - z I

= -^-.U^\-^—\-P{z\ l m z < 0 2m. \ t — z /

where P(z) is a polynomial.

EXERCISES 113

Therefore the solution to (3.87) can be written in the form

g(z) _ 1 /*( / ) 1 *(z) 2iri \k(t)'t-z l m z > 0

^(ζ) = ά(^)'Τ^)' Imz<0

It is assumed that ^ τ is a generalized function £ (Τ>υ>)'. The equation (3.84) can also be transformed to the form G+ —G- = f, as discussed in Chapter 2 and can be solved accordingly.

EXERCISES

1. Let T> be the Schwartz testing function space of C" functions on the real line with compact supports, and let H be the operator of classical Hubert transformation. Show that

Ί) Π Η(Ώ) = {0}.

Hint: Use (JZ/φΧσ) = in sgn(a)(Τφ)(σ); σ R sgn(a) = ( ^

ί ίσ = 0

2. Find the solution to the operator equation Hy = y in 23¿,, p > 1 where / / is the operator of the Hubert transform defined on O'y,. Use your result to show that the solution to the operator equation y = Hy + / is unique in O'u.

3. Let H be the operator of the classical Hubert transformation defined on Lp, p > 1. Show that \\H\\P = v(p), where

f t an ($ ) , \<p*2 VP !«* (£ ) . 2<p<=o

See McLean and Elliott [49]. 4. Prove that the only eigenvalues of the operator H are ±i and that the correspond-

ing eigenfunctions are / + iHf for any / £ 2)[,, p > 1. Are these the only eigenfunctions?

5. Laurent Schwartz has defined the Hubert transform of / G D¿,, p > 1 by ~ i / * C"· (χ)' where * represents the convolution operator. Prove that his definition of the Hubert transform is equivalent to ours. This result has been proved by Carton-Lebrun [21].

4 THE HILBERT TRANSFORM OF SCHWARTZ DISTRIBUTIONS

4.1. INTRODUCTION

Chapter 3 presented the Hubert transform of Schwartz distributions belonging to D',, 1 < p < oo, using an analogue of Parseval's relation. However, this technique does not cover elements of Ί)' that do not belong to "D^. The object of this chapter is to develop the theory of the Hubert transform to the space T)' by using a Parseval-type identity and demonstrate with examples the uses of this method. The technique used in this chapter is essentially the same as that followed by Ehrenpreis and Gel'fand and Shilov in extending the Fourier transform to distributions in T>'. Here is what we are going to do:

We denote by H(D) the space of C°° functions defined on the real line whose every element is the Hubert transform of an element of T>. We equip the space H(D) with the topology transported from T> into H(2?) by means of the operator H, thereby making the operator H defined by

(H/)(x) = lim - / Ά Λ (4.1) (-Ό+ TT J X — t

\t-x\>t

as a homeomorphism from T> onto H(D). The Hubert transform H / of / G. T>' is then defined to be an element of the space H'(T>) given by the relation

<H/, = </,-H<p) V<pEH(D)

With an appropriate interpretation of H(H/), / £ H'(D), we show that

- H 2 / = / V / e C

114

THE TESTING FUNCTION SPACE H(D) AND ITS TOPOLOGY 115

Applications of our results in solving some singular integro-differential equations are also given. In the next section we give an intrinsic definition of the space H(D) and its topology. The method essentially consists in characterizing the space of holomorphic functions defined on the complex plane such that the Hubert transform of its restriction to U is in T), thereby giving also an intrinsic characterization of // '(D).

4.2. THE TESTING FUNCTION SPACE H(D) AND ITS TOPOLOGY

A C°° function φ defined on the real line belongs to the space H(D) iff there exists a function φ G T> such that

φ(χ) = - p.v. / ^ - dt = (Ηφ)(χ) (4.2) ir J-v, x - t

The topology of the space H(D) is the same as that transported from the space T> to H(D) by means of the Hubert transform H. Therefore a sequence {φν}ζ=[ in H(D) converges to zero in H(!D) if and only if the associated sequence {ψν}™=\ converges to zero in Ό, where Hi//„ = <p„ V v G f J.

We now prove some results that will be used in the sequel.

Theorem 1. Let H(D) and Ό^, 1 with support contained in the closed interval [—a, a]. Then by the results proved in Theorem 1 of Chapter 3 we have

1 ra tot*) (H<p)<*> = -P / -21—Λ (4.3)

π J-ax-t

Therefore, as |JC| —» °o,

(ΗφΓ'Οχ) = O In x + a )-°G)

Hence, if «/»(■*) = (Ηφ)(*), then ψ(χ) G DLP. Since ψ is an arbitrary element of H(D), it follows that H(D) C D^. Again, since H is a homeomorphism from DLe onto itself, there exists a θ(χ) G Ό^ρ satisfying

(Ηβ)(*) = φ(χ)

Since Ό is dense in DLP [87, pp. 199-200], there exists a sequence {0„}"=1 in D tending to Θ in T>¿ as v —»oo. Now

* £ f (*(*) - (Ηφν)(ΛΓ)) = | |Η(β«*>-β»)) | | P

*c,||fl«-eí»|l - o as y —» oo

[99 Th. 101, pp. 132-133]. Therefore H(D) is dense in Dt. This completes the proof of part i.

(ii) If {<Pv}"=i is a sequence in H(D) converging to zero in H(D) as v —> oo, then there exists a sequence {ψ„} in T> tending to zero in Ί) as v —► oo SUch that Ηψ„ = φν. Using the properties of H, we have

ΙΚ*ΊΙΡ = Ι|Ηψί?||„ s C, ||ψ<*>||Ρ - 0 as v - , oo (4.4)

this completes the proof of part ii. It can easily be seen that the space H'(D) of ultradistributions consisting of continuous linear functionals on H(D) contains the space (T)Lp)'; that is, H'(Ώ) D (T>Lp)', 1 'u where T>'L, is defined to be the dual space of T)u> and i + i = l . D

43. GENERALIZED HILBERT TRANSFORMATION

We now define the generalized Hubert transformation H / of / E £>' as an ultradis-tribution H / E H'(2?) such that

<H/, φ) = {f, -Ηφ) V<p£ H(D) (4.5)

where Ηφ is the classical Hubert transform defined by (4.1). If g E H'(D), its Hubert transform Hg is defined to be a Schwartz distribution by

the relation

<Η*,φ) = <*,-Η<ρ> V (4.7)

Therefore from (4.7) we have

- H 2 / = / V / E D '

If / E if, \<p< oo, then it is easy to show that the Hubert transform of the regular distribution generated by / is

lim I / ™-Λ t->o+ rr J x — t

\t-X>€

GENERALIZED HILBERT TRANSFORMATION 117

Definition. The derivative of g G H'(D) is defined to be an ultradistribution g' belonging to H'(D) by the relation

(g'.<p) = (g,-<PW) V ^ e H ( D ) (4.8)

Since φ G H(D) implies that φ(1) G H(D), g' defined in (4.8) is a functional on H(D). Linearity of g' is trivial, and the continuity of g' follows by virtue of the fact that H(2?) is closed with respect to differentiation and g G H'(2?). So g' G H'(D).

Theorem 2. Let / G Ό'. Then

(H/)<*> = H( / w ) V t e N

where fik) and (H/)( t ) are the fcth derivatives of / and H/, respectively.

The proof is easy and similar to that given in [70]. For this reason the details are omitted. As in the previous chapter we can easily show that Ηδ = + £ p.v.(j) and H( p.v.( j)) = — ιτδ. Let us now look at some examples that illustrate the applications of our results.

Example 1. Solve the following singular equation in T>':

d-¡¡- + H / ' = Six) (4.9) ax

where /G/ / ' (D) .

Solution. Equation (4.9) can be written

¿ (y + H/) = δ(χ) y + H/ = h(x) + C

or

y = - H / + h(x) + C

This solution is meaningful, since / G Η'(Τ>) implies that Hf G T>'.

Example 2. Find the distribution y G Ό' satisfying the integral equation

y + Hy = f, f G £>' Π H'(D) (4.10)

Solution. Operating both sides of (4.10) by H, we get

Hy + H2y = H /

Hy - y = H /

f-y-y = Hf

or

J-Ιψ- (4.Π, Note that (4.11) is meaningful in 23', since the assumption / e C i l H'(23) implies that H / £ 23'.

4.4. AN INTRINSIC DEFINITION OF THE SPACE H(23) AND ITS TOPOLOGY

In the previous section we saw that the topology of the space H(23) was the one transported from the space 23 onto H(23) by means of the operator H, thereby making H a linear homeomorphism from 23 onto H(23). This simply means that if {ψμ} = ]

is a sequence in H(23) and {φμ} =1 is the corresponding sequence in 23 such that Ηφμ = ψμ for all μ & N \ {0} then Ηπιμ_» ψμ, = 0 in H(23) iff limM_<» φμ = 0 in 23. This way of describing the topology of H(23) is not intrinsic. Even the description of the space H(23) that every element of H(23) is the classical Hilbert transform of some member of 23 is not quite intrinsic. In this section we will consider an intrinsic definition of the space H(23) and its topology.

Definition. The space Ψ. A holomorphic function Φ(ζ) defíned on the complex plane is said to belong to Ψ iff

PI. Φ(ζ) is analytic on C \ [a, b], the closed interval [a, b] varies as Φ varies. P2. Φ(ζ) = O (l / |z | ) as |z| -»°°. P3. lim>_o+ Φ(* ± iy) = Φ±(χ) exists in 23,,, p' = p/(p - 1), 1 <p< t» .

The fact that Ψ is nonempty can be seen by the fact that the function k(z) defíned by

k(z) = / - ^ dt, -»0" , respectively, in 23,/, p' = p/(p - 1), 1 < p < °°, and define

Φ(ζ) = ^ 3 / ~-dt (4.14)

AN INTRINSIC DEFINITION OF THE SPACE H(Z>) AND ITS TOPOLOGY 119

the function ψ(χ) and φ(χ) by the relation

ψ(χ) = Φ + (χ) + Φ~(χ) (4.12)

<p(x) = Φ + (*) - Φ"(χ) (4.13)

Then φ(χ) E H(D), φ(χ) £ D, and

= _L Γ «L 2-m /_«, f -

Moreover

φ(χ) = H ( | ) (JC) (4.15)

/Voo/. Since Φ(ζ) = 0 ( A ) , |ζ| —> °°, the integral defined by

/(c,r) = r—: / A, Imz = y # 0 2ιπ y_„ / - z

exists. Now using Cauchy integral theorem for the upper half-plane, we can show that

1 r <p(/ + ¿6) _ [ φ(ζ + /e), l m z > 0 2 Ϊ Ϊ · ; - ~ ^ 7 " Δ - \ 0 . I m z < 0 ( 4 1 6 )

Letting e —> 0+ in (4.13) and using P3, we get

- ! - / " £ « * = / * « ■ I m z > 0 (4.17) ImJ^t-z \ 0 , l m z < 0 y }

Similarly by performing integration in the lower half-plane and by letting € —» 0~, we can prove that

1 r 0 .. 10. SS/. . 7^7* = 1-*ω- *η*<ο (418)

Combining (4.17) and (4.18) we get,

φ ω = ^ / φ ( / ) ~ φ (f)df, l m z # 0 (4.19) 2ιπ 7_«, f - z

Since Φ(ζ) is analytic outside the closed interval [a, b], it follows that

Φ + (χ) = Φ~(χ) Vx&[a,b]

In other words, <p(x) defined by (4.13) must be zero outside [a,b]. But ψ e Dp-, therefore φ S T>. Using (4.19), we have

Thus

Φ ω = _L f *'X'-*> ώ + ±Γ «<)y dt (4.20) 2 m . / _ . (f - jt)2 + y2 2ir / _ „ (? - Jt)2 + y2

By letting y -+ 0±, respectively, we get

and

<P + W = - i ( H V ) W + i(pU) (4.21)

Φ - (jt) = - i (H<p)U) - J <p(x) (4.22) 2/ 2

Adding (4.21) and (4.22), we get

Φ + (χ) + Φ'(χ)= --(Ηφ)(χ) ι

or in view of (4.12) we get

φ(χ)= - T ( H < P ) ( J C ) E H ( 5 D )

This completes the proof of Theorem 3. D

Corollary. There is a one to one correspondence between the space Ψ and the testing function space H(D).

Proof. We first show that there is one-to-one correspondence between Ψ and T). In view of the result (4.14) of Theorem 3, it follows that for given Φ(ζ) G Ψ, there exists a φ G T) satisfying

Φ ( ζ ) = - ^ Γ - ^ - Λ (4.23) 2m J_x t - z

From (4.23), <p(f) = 0 implies that Φ(ζ) = 0. Therefore Φ(ζ) Φ 0 implies that φ(0 Ψ 0, so the correspondence given by (4.23) is one to one.

Again, if ψ e Ί), then the function 1 ^ ) . Λ ε ,

2 m r »a:

i-a, t -That is, for given we can find a function F(z) in Ψ. Clearly the correspon-dence between Ψ and T> is one to one. But the correspondence between Ό and H(D) is also one to one; therefore the correspondence between Ψ and H(D) is one to one. D

A GEL'FAND-SHILOV TECHNIQUE FOR THE HILBERT TRANSFORM 121

4.5. THE INTRINSIC DEFINITION OF THE SPACE H(D)

In view of Theorem 3 and its corollary, we note that the testing function space H(O) consists of all C°° functions Ψ(χ) of the form — ί(φ+(χ) + φ~(χ)), where Φ(χ + iy) belongs to the space Ψ and the limits <p±(x) are taken in the Ί) pi sense as y —* ±0 (respectively),/?' = (-?-[)·

4.5.1. The Intrinsic Definition of the Topology of H(£>)

We can now define the convergence in H(D) as follows: A sequence {ψμ } = L in H(D) converges to zero in Η(Ό) iff all complex-valued functions φμ(ζ) in Ψ that generate ψμ(χ) in accordance with Theorem 3 are analytic outside a fixed interval [a, b]. Note that this condition enforces that the corresponding function φμ(χ), satisfying

— Ηφμ = ψμ ττι

have their supports contained in the closed bounded interval [a,b\. γ*(ψμ) —> 0 as μ —> oo for all t E N , where γ*(φ) = ||<P(t)|| „ p' = pKp — 1), l 0 in H(D). Therefore the corresponding sequence {<ρμ} =1 in T> converges to zero uniformly on every compact subset of R [87, p. 200] along with all its derivatives. Likewise {<ρμ} =1 converges in Ό. That is, if Ηφμ = φν, then ψ„ —♦ 0 in H(D) implies that <ρμ —+ 0 in T>. The fact that {<ρμ } = converges to zero in T) implies that φ„ —> 0 in H(D) is trivial to prove. So the mode of convergence of the sequence { ψμ } = { in H(D) to zero can be intrinsically defined by Theorem 1.

4.6. A GEL'FAND-SHILOV TECHNIQUE FOR THE HILBERT TRANSFORM

Gel'fand and Shilov in [44, pp. 151-154] have extended the Hubert transform to generalized functions of the space Φ' and have proved the inversion formula -H2f = f for all / G Φ' where elements of the testing function space Φ belong to the Fourier transform space of a testing function space Ψ (i.e., Φ = ?(Ψ)) where Ψ is equipped with the topology generated by a countable set of norms and the topology of Φ is that transported from the space Ψ by the operator of Fourier transformation J. They claimed that a regular distribution generated by a locally integrable function f(x) satisfying the asymptotic order of f(x) = 0(\x\l~f), x —> ±°°, 0 < e < 1, is Hubert transformable according to their scheme. As it stands, their claim is

incorrect. Nevertheless, their technique is ingeneous and has helped us to extend the Hilbert transform to a space of ultradistributions containing the space S' of tempered distributions. We will consider this extension of the Gel'fand-Shilov technique in the next section. The technique of Gel'fand and Shilov given below is a simplified version of their technique discussed in [44, pp. 151-154].

4.6.1. Gel'fand-Shilov Testing Function Space Ψ

The testing function space Ψ consists of functions i/>(s) (—oo < j < oo) possessing the following properties:

1. The functions ί*ψ(ί) is absolutely integrable on the line — °° < s < °° for any k = 0,1,2,3 and i¡>'(s) is bounded on the real line.

2. ψ(ί) is continuous and has a continuous derivative ψ'(ί) on each of the half-lines -oo < j < 0, 0 ^ J < oo; the functions i¡t(s) and i/»'(s) may have a discontinuity of the first kind at the point s = 0.

3. J V ( S ) is absolutely integrable on the line -oo < s < oo for any k = 0,1,2,3

It is now a simple exercise to show that

max |ψ(ί)| < oo —oo<j<oo

Gel'fand and Shilov introduce a topology on the space Ψ by introducing the following countable set of norms:

H(s)\\k = / \¿Ws)\ds+ í | i V ( j ) | & + max |ψ(*)| 7 - » 7 -00 -oo<,<oo

+ max \é'(s)\ [110, pp. 7-21] -oo<j<oo

With this topology the space Ψ is a complete countable multinormed space. They have used an ingeneous technique to show that the Fourier transform of each ψ ε Ψ is infinitely differentiable and satisfies the asymptotic order ?(Ψ)(χ) = O (fa), x —» ±oo. Another elementary proof of this fact can be given as follows:

Therefore

(7Ψ)(χ)= I t(s)eisxdx J — 00

/ | - ( ^ ( i ) c ' " )d5= / isMs)eis*ds 7-00 OX 7-00

I y ¿(*(V")&|s / \sm\ds

A GEL'FAND-SHILOV TECHNIQUE FOR THE HILBERT TRANSFORM 123

and

f_^Hs)eÍSX)dS

is uniformly convergent. Hence using the standard result on the switch in the order of integration and differention, one can show that

^ ( J * X * ) = / ms)seisxds

Using similar argument and the method of induction, one can show that

k

(έ ) ( : w w = / i"*** "* Clearly φ(χ) = (ΤΦ)(Χ) is infinitely differentiable on the real line.

We now prove that (Τφ(χ)) = O (fa), x -> ±00. We first show that ϋπ^-,+οο ψ(ί) = 0.

For a > 0 we have

wv Φ(Ν) = φ(α) + / i¡i\s)ds

Ja

for ψ(ί) is absolutely integrable in the real time (—°°, °°). Letting N —► 00, we have

lim φ(Ν) = φ(α) + [ ψ'(ϊ)Λ W-» Ja

Therefore limf_a, φ(ί) exists. Again, since fa |)//(J)| ds exists and l im^» ψ(ί) exists, it follows that lim,-,«, i//(s) = 0. Similarly it can be shown that lim,-,-«, I/Í(Í) = 0. Using the technique of integration by parts for improper integrals, we get

σΦ^^Γφ^^ + ^ - * ^ IX

\(?φ)(χ)\ < ~ [ |ψ'(*)|ώ + τ^ \φ(0-) - ψ(0+)| \x\ J-<» 1*1

Hence

(ΙΦ)(Χ) = O (±λ , X - ±00.

If i//(s) e Ψ, then φ^-s) also belong to Ψ:

(Τ(Φ(-ί)))(χ) = / φ{-ί)ε+"*άί = / Μ)β-"*Α = <p(-x) y —00 y —00

Therefore, if <p(t) G. Φ, then <p(—t) e Φ, and we have

?(Ψ) = Φ and ?(Φ) = Ψ

In other words, Φ and Ψ are Fourier transforms of each other. We now wish to show that Φ is closed with respect to the operation of the Hubert

transformation H. Let ψ £ Φ, with Τ(Ηφ) = i sgn(;c)( J<p)(x). Note that Ψ is closed with respect to multiplication by sgn(jc). Therefore

Η Φ Ε Φ

since Φ and Ψ are duals of each other. This completes the proof of the fact that Φ is closed with respect to the operation of the classical Hubert transformation H.

4.6.2. The Topology of the Space Φ

The topology of the space Φ is the one transported from the space Ψ by means of the operation of the Fourier transformation. A sequence {«M"=i (where <pv = Jtyv) converges to the identically zero function as v —» °° in the topology of the space Ψ iff ll«M* ->0as v -> oo for each * = 0,1,2,3

It will be an interesting problem to define the space Φ and its topology in an intrinsic way. Gel'fand and Shilov have given an easy proof of the inversion formula

Η2φ = -φ V(p e Φ

as follows: Given

Τ(Ηψ) = Í sgn(jr)Τφ a.e. V φ G Φ

Now Ηφ e Φ for all φ G Φ. Therefore

Τ(Η2φ) = 7(Η(Ηφ)) = isgn(je)J(l/9)

= íSgn(jr)/sgn(jr)CF<p)

= -J<p a.e.

Then

Τ{Η2φ + φ) = 0

and//2<p = - φ every where V φ £ Φ, since <p,//ip,//2<pareall continuous functions. If / e φ', its Hubert transform H*f was defined by Gel'fand and Shilov [44] as an element of Φ' satisfying the following relation:

< / / * / » = </,// ν < ρ £ Φ

where Ηφ is the classical Hubert transformation of the function φ.

AN EXTENSION OF THE GEL'FAND-SHILOV TECHNIQUE 125

It is easy to show that

(Hmff = -f ν / ε Φ '

since

(Η'(Η'/),φ) = (Η*/,Ηφ)

= <f.H2q>)

= (f, ~φ) ν φ Ε Φ

Therefore

H*2f = -f ν / ε Φ '

They claimed that a function / that is locally integrable and satisfies the asymptotic order f{x) = 0[xl~e], x —» ± » forO < e < 1, is Hilbert transformable according to their definition [44, pp. 153-154]. As it stands, this claim is wrong, for the integral f™x f(x)(H<p)(x) dx is divergent for all nonzero ψ G Φ. We will now extend the Hilbert transform to a space of generalized functions that is closed with respect to multiplication by polynomial functions and that also contains the regular generalized function generated by locally integrable functions f(x) satisfying the asymptotic order/U) = 0(jc l - e) , x - x » , 0 < e < 1.

4.7. AN EXTENSION OF THE GEL'FAND-SHILOV TECHNIQUE FOR THE HILBERT TRANSFORM

In this section we have extended the technique of Gel'fand and Shilov for the Hilbert transform to a space of the ultradistributions, where multiplication by a polynomial is permissible. Thus in this section we will be able to solve linear singular differential equations with polynomial coefficients by using the Hilbert transform technique developed. Our extension technique will be based upon the classical relation

{(ΤΗ)φ)(x) = i sgn(x)(J<p)(x) V φ G S

Using Parseval's relation for Fourier transform, we have

(?Ηφ)(χ)ψ{χ)άχ= / (Ηφ)(χ)(Τφ)(χ)άχ ν φ , ψ 6 5 -00 J — 00

/

OO - 0 0

isgn(x){J<p)(xW(x)dx = / (Ηφ){χ)(Τψ)(χ)άχ 00 J -00

Dividing through by / and using Parseval's relation for Fourier transform, again we have

<p(x)J (sgn(xMx)) dx = -i / (Ηφ)(χ)(?ψ)(χ)άχ •00 J —00

In the duality notations this result can be rewritten as

(φ, J ( sgn^( jc ) ) ) = (Ηφ, -σ(ψ(χ))) V ψ, ψ G 5 (4.24)

where the operators J and H are defined as follows:

J(<p)(x)= j <p(t)e+itxdt V 0+ J X — t

\x-l\X

The relation (4.24) will be important to us. By using the analogue of the relation (4.24), we will extend the Hubert transform H and its inverse H~l to a certain space of ultradistributions.

4.7.1. The Testing Function Space St

The set S\ is defined to be the set of C°° functions defined on U satisfying Si C S, where S is the testing function space of Schwartz tempered distributions such that each φ G Si is zero in an open interval containing the origin. The topology of Sj is that induced on it by S. Clearly Si is a locally convex, Hausdorff topological vector space but is not sequentially complete. The space Si is not dense in S either.

4.7.2. The Testing Function Space Zx

An infinitely differentiable complex-valued function ψ defined on the real line is said to belong to the space Z\ iff it is the Fourier transform of an element ψ G Si. Since the operator J of the Fourier transformation is a homeomorphism from S onto itself, it follows that Z\ is a subspace of S. We equip Z\ with the topology induced upon it by the topology of S. The space Z\ is also a locally convex Hausdorff topological vector space but is not sequentially complete.

We now will show in brief that Z\ is not dense in S. Assume that it is so. Then

Zi =S

JXS,) = S

JXS,) = S

Si = J~\S) = S (S, is dense in S)

which is a contradiction. In that case Theorem 1 as stated and proved in [76] is wrong.* However, the convergence of a sequence to zero in Z\ also implies the convergence of the same sequence to zero in the space S. Therefore the restriction of / G S' to Z\ is in Z,\ but there may not be one-to-one correspondence between Z[ and S'. However, since any tempered distribution is a continuous linear functional on Z\, we can say that Z[ D S'. Thus any distribution of compact support is an ultradistribution in Z[.

4.7.3. The Hubert Transform of Ultradistributions in Z',

In analogy to the relation (4.24), we now define the Hubert transform H/ of / G Z[ by

(H/, - i j (9 )> = </. J{sgn(x)<p(x))) V φ G 5, (4.27)

The relation (4.27) defines H/ as a continuous linear functional over the testing function space Z\. In fact H/, the Hubert transform of / G Z[, is the element of Z[ that assigns the same number to —ϊ?(φ) as / assigns to J"(sgn(;c)<p(x)) for all

ψ G 5, (4.28)

The inversion formula — H2/ = / can be proved as follows:

(H2/, -ij(<p)) = (H(H/), -iJ(<P)>

= (H/, J(sgn(jc)<p))

■Η*( ))> = (/.&&)) V<pG5

= < - / . - i ' J ( v ) )

Therefore

H 2 / = - / V / G Z,' (4.29)

Note that our technique has the advantage that (4.29) is derived without using the corresponding classical inversion formula. In Example 5 we will show that H(t"f)(x) = x"(Hf)(x) for all n G N and / G Z{. By this fact it can be shown that the Hubert transform of the ultradistribution generated by a polynomial p(t) is a zero ultradistribution, a fact that can also be verified by manipulations. Since more general results will be proved in the next chapter, the proofs of these facts are

"This error was pointed out by C. Carton-Lebrun.

being omitted here. It now follows that the inversion formula (4.29) establishes the uniqueness theorem for the operator H in Z[ modulo the space of ultradistributions whose Hubert transform is zero. The class of such distributions will be called the zero ultradistribution. Unlike the other spaces of the Hubert transformable generalized functions, it is harder to prove some of the basic formulas in the space Z[. I give some examples below.

Example3. Show that (H8)(x) = (1/·π-)ρ.ν.(1 /x) when δ G Z[.

Solution. By definition (4.27), we have

(Ηδ, -Ϊ7(φ)) = (8, J(sgn(jr)v)) (4.30)

sgn(x)<p(x)e+ix' dx - ( δ < " · / .

-F sgn(jr)i>(jt) Ac (4.31)

Now

^ Ρ . ν Λ ^ ) ) = (^(^ρ.ν.1),φ)

= (-sgnU),<p(*))

= f - sgnOOrfx) dx (4.32) J —00

From (4.31) and (4.32), we have

/ l ρ.νΛ -&(φ)\ = (m, m<p)) v Ψ e s,

namely Ηδ = (1/ττ)ρ.ν.(ΐΛ) inZ/. Further generalization of the spaces Z\, Z[ will be discussed in the next chapter.

The spaces Z\ and Si are closed with respect to differentiation. Thus the distributional differentiation on Z[ can be defined as

(D/, 7(9)) = </. -DJ(*)> V φ S S, (4.33)

So, if / e Z[, then Of £ Z[.

Example 4. Show that (H/) ' = H / ' for all / G Z[.

Solution.

(Hf',-if(<p(x))) = (/'.J(sgn(jr)<p(jc)))

= ^ / . - ^ J " ( sgnW<pW)^

= (f,-T(+ixsgn(x)<p(x)))

= (/,-j(i*sgn(A:)<p(jc)))

= (H/ ,¿J(W(P(A:)) )

= (H/,J-(-x<pW))

= (Η/,^ΐ(φ(χ)ή

= (jt"f>-&(**)))

That is,

<H/', -ΐΤ(φ(χ)) ) = <(H/)', - /5(φ(*)) ) V φ £ S,

Therefore H / ' = (H/)'. Using induction, we can show that H/<*> = (H/)<*>.

Example 5. For / £ Z[ show that

H(/m/) = . A H / X J C ) V m £ N (4.34)

Solution. We will prove the result (4.34) for w = 1, since the general result can be proved by induction:

(H(tf),-ij(<p(x))) = (tf,j(sgn(x)<p(x)))

= </,ij(sgn(jcM;c)))

= </. J(+«'sgn(Jc)v'(x))>

= < H / , - I J ( + IV(JC)))

= (H/, -ίχΤ{φ))

= (JCH/. -iT(<p))

Thus

(Η(ί/), -ij(<p)) = (*H/, -/J(<p)) V φ £ S,


Therefore

HO/XJO = *H/ v / e z ;

Remark. It may be noted that the spaces S\ and Z\ lack the completeness property with respect to the topology induced on them by S. However, by modifying their definitions slightly, we can define new testing function spaces Sv and Ζη as follows: For η > 0, we define the testing function space 5,, as the space of C°° functions defined on IR such that S,, C S each φ e 5η vanishes in the open interval ( - η , η). The space Zv is defined by

zv = nsv) The spaces Sn and Zv both are subspaces of S, and both are sequentially complete. As such, ST, and Ζη both are Frechet spaces. Duals of S,, and Ζη will be denoted by 5η and Ζ'ψ respectively, and all the preceeding results and examples are valid for S'v and Z^ too.

While dealing with some specific problems such as boundary-value problems, it is a useful idea to have a testing function space which is a Frechet space.

To this end we can also construct the testing function space So as the strict inductive limit of the sequence {S,,,}^, where ηι > ifc > "' and lim,-,«, η, = 0. That is, we define S0 = \J¡Li vr Since S,,, C S^ C S^ · · · and the topology of Sm is stronger than that induced on 5 m by S,,i+1 for each k = 1,2 we can say that a sequence {φν)Ζ= i i" So converges to zero in So 'ff

1. φ„ €Ξ SVk for some k and all v G N. 2. φν —y 0 as v —* °° in S^.

Clearly the convergence of the sequence to zero in SVt implies convergence of the same sequence to zero in S^t+, and the convergence to zero in Sm+2, and so on.

Example 6. Solve in Ζ'ψ η > 0,

xy + Hy = / ' where / £ Z¿ (4.35)

Solution. Operating on both sides of (4.35) with H, we get

U(xy) + H2y = H / '

xHy - y = (H/) '

x[f' -xy]-y = (H/) '

(x2 + l)y = xf - (H/) '

or

DISTRIBUTIONAL HILBERT TRANSFORMS IN «-DIMENSIONS 131

4.8. DISTRIBUTIONAL HILBERT TRANSFORMS IN /»-DIMENSIONS

Our objective in this section is to extend the Hubert transform and its inversion formula along with many distributional results to n-dimensions. The theory of the Λΐ-dimensional Hubert transform, in general, turns out to be very complicated. Since our definition of the «-dimensional Hubert transform will be expressed in terms of the properties of the corresponding «-dimensional Fourier transform, our results turn out to be quite easy to obtain. A more efficient and the organized work on the n-dimensional Hubert transform will be presented in the next chapter. I now give some definitions and preliminaries.

4.8.1. The Testing Function Space S,(R")

The testing function space 5i(Rn) is defined to be the space of complex-valued infinitely differentiable functions defined on R" such that Si(R") C S(W), and each φ E S\ (Rn) vanishes in an open ball centred at the origin (the ball may depend on φ). The topology of Sx(W) is the same as that induced on Sx(W) by S(R"). Clearly Si(R") is a locally convex Hausdorff topological vector space but is not sequentially complete. The dual of SX(U") is denoted by S[(W).

4.8.2. The Testing Function Space Zi(W)

The testing function space Z\ (R") is defined to be the space of infinitely differentiable complex-valued functions on R" such that

Z,(R") = j(S,(R"))

That is,

z,(R") = {φεs(R»): φ(χ) = jew*)νψεsdW)}

where J is the Fourier transform operation in R". Since J is a homeomorphism from S(W) onto itself it follows that

Z,(R") C S(W)

The space Zi(W) is also a locally convex Hausdorff topological vector space but it is not sequentially complete. The restriction of / e S'(R") to Zi(Rn) is in Z,'(R"). It is in this sense that we write

Z,'(R") D S\W)

but there is not one to one correspondence between Z{(W) and S'(W). In other words, for an element in Z^R") there may be more than one element of S'(U.n) that may correspond, since the space Z^R") is not dense in S(W). This fact can be easily proved by reasoning similar to that given in Chapter 2.

The completeness of the testing function space is considered to be quite an im-portant property especially when we have to deal with partial differential equation problems. We therefore construct the following testing function space.

4.8.3. The Testing Function Space SV(W)

The testing function space 5η(Κπ) consists of infinitely differentiable complex-valued functions defined on IR" such that every element of 5^(R") belongs to S(W) and vanishes in the region \χχ\ < η, |JC2| < η U„| < η. The topology of S^(R") is the same as that induced by 5(Rn). It can easily be seen that S^(R") is a sequentially complete locally convex Hausdorff topological vector space. Clearly S^R") C S(W) and the restriction of an element of / e S\W) to SV(U") is in S'V(R").

4.8.4. The Testing Function Space Z^W)

The testing function space ZV(W) is defined to be the space of infinitely differentiable complex-valued functions on W such that for every φ G ZV(U") there exists a φ Ε SV(R") satisfying

<PW = (JVOto

That is,

Zr,W) = f(Sv(W))

where J is the Fourier transform operator on W. The space Ζη(Κ") C S(U") and the topology of Ζη(Μ") is the same as that induced on ZV(W) by S(W). Ζη(Κ") is clearly a Frechet space. The restrictions of / e S'(W) to Z^W) is in Z^W).

4.8.5. The Strict Inductive Limit Topology of ZV(W)

Let {T7,-}JL , be a sequence of diminishing positive numbers tending to zero. Then clearly

£"<\\ - ¿m ^- ¿m C · · ·

The topology of Zn is stronger than the topology induced on Ζη, by Ζη/+1. We must now define the countable union space Zo by

00

Zo = \JZVi (4.36) ( = 1

A sequence {<p„}"=1 belonging to ZQ(U") is said to converge to zero in Zo(R") iff it belongs to some ZVi and converges to zero as v —► °° in the topology of Ζη/. The topology defined in this way on ZQ is said to be the strict inductive topology of the sequence ZVi{W). The elements of Z¿(W) will be called ultradistributions. We

DISTRIBUTIONAL HILBERT TRANSFORMS IN n-DIMENSIONS 133

say that a sequence {φν}™=ι is a Cauchy sequence in Zo(W) iff for all v £ W \ {0}, ψν £ ZV¡(W) for some /' £ ^J, and {tpm(x) - φη{χ)} tends to zero in ZVi as m and n —» oo independently of each other. It is easy to see that Z¿(W) is a sequentially complete locally convex Hausdorff topological vector space.

Analogously we define the space

SO = (jSr,,

The topology, the completeness concept, and the convergence of a sequence in So can be defined in the same way as in Zo- We now define the Hubert transform H / of an uhradistribution / £ Z¿(U") as an ultradistribution on ZQ(IR") by the relation

(H/, -ϋ{φ)) = (/, j(sgnU)<p)) V φ £ SoiW) (4.37)

It is quite evident from (4.37) that H/ defined by (4.37) is a linear functional on Zo(W). If {φ„}"=, is a sequence -> 0 in S0(W), then J,(sgnU)<p„) -> 0 in Zo(R") as v — oo. Therefore H / £ Z¿(W).

Theorem 4. Let Z¿(W) be the space of ultradistribution defined in Section 3.5, let ^ ( 0 = Eosi/is* ai''> w h e r e

and let

ΐ'·ι = Σ ό

be a ¿-degree (finite) polynomial. Then there exists an ultradistribution belonging to Z¿(R") that corresponds to this polynomial and the Hubert transform of the ultradis-tribution generated by Pk(t) is zero.

Proof. 1. We generate an ultradistribution from the polynomial Pk(t) using the relation

(Pat), φ) = J Pk(t)<p(t) dt V ψ £ Zo(R") (4.38)

The integral in the right-hand side of (4.38) clearly exists as φ £ Zo(U"). The linearity of the functional Pk(t) defined by (4.38) is trivial. Continuity of the functional /**(*) can as well be proved by using the properties of S(R").

2. For P/dt) = tk, where k is an arbitrary Λ-tuple with each component in N, the general result can be proved by applying the result

(Hf- . iJ to))

= (tm, j(sgnW<p)) V φ e S0(W) (4.39)

= Itmj(sgn(x)V)(t)dt R"

= I dttm Í sgn(x)<p(x)e+'x'dx R" R"

N N N

= lirn^ i Í ■■■ idt(i)m /sgn(x)<p(m)(x)e+ix'dx -N -N -N R»

N—<° J X\ Xl R"

sin(Mrm) (by switching the order . . . . . ¿* /■ . .· λ (4.40)

xm of integration) It can be easily proved that the right-hand-side expression in (4.40) tends to zero as N —» °°. □

We now state some results without proof that can be proved quite easily.

For δ G Z¿(ñn)

(H8X,) = <!Wru<üp.v. π" \xj π"

1 X\*2 ' ' ' χη.

These symbols will be explained in greater detail in the next chapter.

Hfm) = (H/) ( m ) V in = 0 . 1 , 2 , 3 . . . .

where

</(m)- ψ) = ( - i)(m)</. <P(M)> v φ e ZodR")

Wmf)(x) = xm(Hf)(x)

where

#m — f/rti fnt2 . . . tmn 1 ~ 'l «2 'n

DISTRIBUTIONAL HILBERT TRANSFORMS IN «-DIMENSIONS 135

Note that

S(x) i + UP e z¿(W)

since Zo(W) is not closed with respect to multiplication by (1/1 + \x\2). But (1 + x2)f e Z¿(W) if / £ Z¿(W) because Zo(W) is closed with respect to multiplication by polynomial functions.

Example 7. Let us find the Hubert transform of

(1 + x2)8(x) E Z¿(Un)

Solution. Customarily we have

(1 + χ2)δ(χ) = (1 + 02)δ(χ) = δ(χ)

Therefore

H [(1 + χ2)δ(χ)} = Ηδ = — p.v

But, if we make use of the formulas derived in this section, we get

H [(1 + χ2)δ(χ)] = (1 + ί2)Ηδ = (1 + l2)^ p.v. j

We got an additional term like

1 1 I — p.v.-

By a method analogous to that used in Theorem 4, we can show that

1 1 t t — p.v.- = —

π" t π" (ultradistribution)

By using a method similar to one used in proving Theorem 6, one can show that the ultradistribution generated by a polynomial is zero ultradistribution. Therefore t/ττ" represents a zero ultradistribution. Hence

Η[(1+*2)δ(χ)] = ^ ρ . ν .

The inversion formula (4.29) can be extended in Z'(W) as

H 2 / = ( - ! ) " / V / e Z ' ( R " )

EXERCISES

1. (a) Prove that f(x) = x defined on the real line is absolutely continuous on the real line.

(b) Prove that an absolutely continuous function f(x) on the real line is uniformly continuous there but is not bounded on the real line. Suggestion: Try the counterexample f(x) = x.

2. Prove that a function /(JC) defined as

( sin*

7' sin*

fix) x = 0

is a C°° function on the real line and that the function f(x) does not belong to the testing function space Φ, even though f(x) and each of its derivatives is

o(A).w —. Hint: Prove that the function g(x) defined by

1 f sin/ _, · ,_, g(x) = = - / - — e dt

is discontinuous at the points x = ± 1. It is easy to see that

I f 2sinf g(x) = -— / cos tx dt 4π J^ t

= I Γ [ s i " { ( l + ^ } + s i n { ( l - * ) / } ] i f e t c

IT 7-00 /

3. Prove that the function f(x) = e" w , — <» < χ < oo belongs to the testing function space Ψ. Prove that the function g(x) defined by

( e-(\/x2)-x2

sw = \: · ' * ° x = 0

also belongs to the testing function space Ψ. 4. Let D be the Schwartz testing function space consisting of C°° functions with

compact supports. Let Q be the space of functions obtained by multiplying each element of D by sgn(jc) where

(\ if JC > 0 sgn(x) = < 0 if x = 0

l - l if JC < 0

That is, Q = {/(*); f(x) = sgn(x)<p(x), φ G D}. Prove that Q C Ψ.

EXERCISES 137

Prove that

{fix); /(*) = sgn(jr)?(jc). φ(χ) E S} C Ψ

where 5 is Schwartz testing function space of rapid descent. 5. Give an intrinsic definition of the space Φ. Hint: Let φ(χ) be a C°° function defined

on the real line such that

»'"«>=°(H) :°°, * = 0,1,2,3, . . .

Here <p(0)(*) = <p(x). Add some more conditions to φ to prove that ψ(χ) G Φ:

(J~'(?))(*) e *

6. Give an intrinsic definition of the topology of the space Φ (open problem). 7. Let φ Ε S0(R). Prove that

lim / smnx „ „ , ^ ,

φ(χ) sgn(*) a* = 0

Note that l im^» ^ = δ(χ) in D'. [See Zemanian [109].] 8. Gel'fand and Shilov defined the space Ψ of all functions ψ($) (—oo < s < oo),

possessing the following properties: (i) The function /if/is) is absolutely continuous on the line — °° < s < °° for all

k = 0,1,2,3 (ii) ψ(ε) is continuous and has a continuous derivative ψ'(ί) on each of the half-

lines —oo < s s 0, 0 ^ J < oo; the function i¡/(s) and ψ'(ί) may have a discontinuity of the first kind at the point s = 0.

(iii) ί*ψ'(5) is absolutely integrable on the line — oo < s < oo for each /t = 0,1,2,3

Prove or disprove that (a) sup |ψ(ί)| < oo

— oo<5<oo

(b) sup |ψ'(5)| < oo -oo<j<oo

(c) Prove that this space, Ψ defined by Gel'fand and Shilov, are different from that defined in this chapter. Illustrate this fact by giving a specific example.

5 «-DIMENSIONAL HILBERT TRANSFORM

5.1. GENERALIZED π-DIMENSIONAL HILBERT TRANSFORM AND APPLICATIONS

This chapter extends the distributional Hubert transform H and the corresponding inversion formula — H2 = / t o «-dimensions. It derives many related results and demonstrates their applications in finding solutions to some singular integral equa-tions in the space T>'P.

5.1.1. Notation and Preliminaries

As usual we denote by IR" and C the real and complex «-dimensional Euclidean spaces, respectively. An element of IR" and C" is an «-tuple x = (χχ,χι,... ,χ„) and z = (Z|, Ζ2,..., z„), respectively, z; = x¡ + iy¡ (j = 1,2 ri). We also write

i «, j a |a| = a, + a2 + · · · + an dX\a* . ..dx„a"

„a _ a, a2 X — X\ X-¿ ,

5.1.2. The Testing Function Space O^iW)

An infinitely differentiable function φ defined over W is said to belong to the space T>t(W) iff Da φ(χ) belongs to LP(W) for each \a\ = 0 ,1 ,2 ,3 , . . . ; we introduce a topology on ©^(R'1) with the help of the sequence of seminorms ym on ©^(R") as

138



GENERALIZED n-DIMENSIONAL HILBERT TRANSFORM AND APPLICATIONS 139

follows: For each φ 6 DLP(R") we define

UP

(5.1) ?I«|(<P)= [j\Da<p{x)\pdx

where \a\ = 0,1,2,3 Since γο is a norm the collection of seminorms ym is separating [110, p. 8]. Therefore the space D^R") is a countably multinormed space. We denote by V'AR") the dual of the space T> ,<(Rn) where p' = p/(p - 1), p > 1. We say that a sequence {φμ} =1 converges to φ in D^R") if, for each \m\ = 0,1,2

Ύ\η,\(<Ρμ - φ)->0 as μ -» °°

The topology of D[p(R") is generated by the seminorms 7|m| in the usual manner [110, pp. 8-14]. The Schwartz testing function space TKRn) is dense in D^R") [101]. We state here the structure formula for / £ D^R") , which can be proved in view of similar results proved for spaces K(MP) [44, pp. 109-110]. See also [87, p. 201].

Theorem 1. Structure Formula. If / G Ό'^η"), 1 < p < °°, then / is equal to the finite linear combination of the derivatives of function in Lp(Rn). That is, for each / G D^(Rn) there exist /„'s in lf(U"), satisfying

* f </- Ψ) = Σ (~1 ) M / D" <**)/«Wdx V ? e ^Α^") (5.2)

M=0 B„

5.1.3. The Test Space *(R")

The space X(M") consists of all test functions φ(χχ, x2,..., x„) on R" of the form k

Ψ(Χ) = ^ tu, (*1 )ψμ.2(χ2) ■ ■ ■ ΨμΑχη) (5-3) μ = \

where <fy,U/) G !D(R) Vy = 1,2,...,« and \/μ = 1,2,3,..., k, where k is some finite number and μ; are positive integers ^ fc such that μ, # μ, if / Φ j . We consider the relative topology on X(U") induced by the Schwartz testing function space TKU").

Lemma 1. The test space X(R") is dense in the space ΊΧΜ").

Proof. Let φ(χ) G D(R") with support contained inß(a) = {x G R" : \x¡\ < a, 1 < /' ^ «}, a > 0. By Weierstrass' approximation theorem [106, pp. 68-71], for an integer m > 0 there exists a polynomial Pm(x\ ,x2,...,x„) such that in B(2a) = {x G R" : |JC,| < 2a, 1 ^ ιr ^ n}, it differs from <P(JCI,JC2 *„) by less than ¿ together with all its derivatives up to order m.

140 «■DIMENSIONAL HILBERT TRANSFORM

Thus

\<pm(x) - /»«(*, ,x2, x3 *„)| < - (5.4) m

V|*| =0 ,1 ,2 / n a n d U , | < 2 a

Since φ(χ) = 0 outside U,| > a, i = 1,2,3, n, it follows that

\P£Xx\.xi * « ) l < - (5.5) m

for

a < \x¡\ <2α , i = 1,2,3 «

and |ít| = 0,1,2 m. Now, let e(x) G. ΊΧΚ), such that e(x) = 1 for |*| < a and zero for |JC| ^ 2a such

that 0 < e(x) ^ 1 for all x G R. Then

{/>„,(*)£(*)£=, (5.6)

where E(x) = ¿(xOefe)' · · e{x„) converges to <p(x) in T> and it can be easily noted that

Pm(x)E(x) E X(W) for each m G N

Since each element of this sequence and the functions φ(χ) have their supports contained in\x¡\ ^2a,i = 1,2 n, we need only show that the sequence converges uniformly to φ along with all its derivatives to the corresponding derivative of φ in the region \x,| s 2a, i = 1,2,..., n. Now

\<p(x) - Pm(x)E(x)\ = |φ(χ) - Pm(x) + Pm(x)(l - E(x))\

< \<p(x) - Pm(x)\ + \Pm(x)\ 11 - E(x)\ (5.7) 1 1

< — + -m m

This is because 1 — E(x) vanishes in the region |jc,-| ^ a, i = 1,2,3 n and 1 - E{x) < 1 in the region a < \x¡\ < 2α,ι = 1,2,3 n.

Therefore, letting m —* <*> in (5.7), we can prove that the sequence {Pm(x)E(x)}2=i tends to <p(x) uniformly on W.

Next consider the sequence of the first derivative:

|φ(1)(*) - D{Pm(x)E(x)}\

= \<pm(x) - P„m(x)\ + \Pmm(x)(l - E(x))\ + \Pm°\x)DE(x)\

1 1 A ^ l· — H »0 as m —► »

/M m m

GENERALIZED /»-DIMENSIONAL HILBERT TRANSFORM AND APPLICATIONS 141

for every x satisfying \x¡\ < 2a, i = 1,2 n, where

A = max —E(x) l'=l....,n dX¡ xeR" '

Therefore the first derivative of the sequence (5.6) converges uniformly to the corresponding first derivative of φ as m —» oo. Carrying similar argument and using the techniques of induction we can show that the sequence (5.6) converges to φ in T> as m —► oo. D

Corollary 1. The space X(U") is dense in the space D^R") .

Proof. Let φ G. D^R") . Since TKU") is dense in Dlf(R'1) [87], there exists a sequence {ψμ} =1 in 2?(R") such that

DL(.(R") ψμ > φ as μ —> oo (5.8)

Since X(R") is dense in D(Rn) by Lemma 1, for each fixed μ there exists φμ,„ £ X(R") such that

φμ,κ ► ψμ as v —> oo (5.9)

Since the convergence of a sequence in D(R") implies its convergence in O^(W) [87], it follows that

2V(R") , (Ρμ,μ ——♦ ψμ a s 1/ —► 00.

We now have the following situation:

«Pi.i- -»· Ψι

Ψΐ.ν -» Ψ2

<Ρμ. -+ Ψμ

and ψμ ► φ. Since the space 25,» (Rn) is metrizable with metric given by

ρ(φ, Ψ) = Σ 1 Ύη>(ψ - Ψ) L-r 2 - 1 + γ„(φ - ψ)

ΛΊ — U

Therefore, by using the result [88, p. 8] there exists a subsequence of {φμ,„}°° v=i

that converges to φ. Therefore X(W) is dense in 25,» (RB). D

142 π-DIMENSIONAL HILBERT TRANSFORM

5.2. THE HILBERT TRANSFORM OF A TEST FUNCTION IN X(W)

Let

k n

φ(χ) = 5Z <V Π Ψμ(Χί) μ=1 i= l

where each of ψμ,{χΐ) belongs to D(IR). Then the function φ belongs to X(R"), and conversely, only a function belonging to the space X(W) has this representation.

We dehne the «-dimensional Hubert transform of φ as

(Ηφ)(χ) = lim — / , f f -2Q— dt ¡=1,2 n ¡=1

Σ!=,^Π;=Ι<ΡΜ('.)

k n

μ=1 1=1

|(,-ΛΓιΙ>ί,>0 ι = 1,2 π

f l im(l /W) / *<*>] «,-.0+ J (X¡ - t¡)

L | / ,-jr,|>í,>0 J

(5.10)

(H<p)(jc) = hm — / — ^ — F i—dt (5.11)

(5.12)

= Σ>Α1Π(Η,φμ(ί,))(*,) μ= 1 l= 1

(5.13)

Each of the terms (Η,φμ)(χ,) e Ό^,ρ > 1. It follows that (H<p)(x) given by (5.13) belongs to Ό^(IR"), p > 1, whenever φ e X(R").

Applying the operator H on both sides of (5.13), we get

It follows that

(Η2φ)(ί) = ΣεμΥΙ(Ηί2)φΙί(ίί)

μ = 1 / = 1

* n

= (-írj^ijívfo) = (-i)V(o μ=1 ,= 1

Η 2 = ( - 1 ) " / inA^R") (5.14)

We now prove that the operator

H : X(W) -> VLP(W), p > 1

is a bounded linear mapping. Linearity of H is trivial. The boundedness property of H is a special case of a more general result [99].

THE HILBERT TRANSFORM OF A TEST FUNCTION IN X(W) 143

Theorem 2. Let H be the mapping defined by (5.12). Then H is a linear continuous injection of X(W) into T)t(W).

Since X(U") is dense in T>t(Rn), the continuous injection H : X(W) -+ D^R") extends uniquely to the whole space "DîU") by the limiting process as follows: For ψ G DJPÍW), there exists a sequence {φμ} = 1 in X(U") such that

DL,(R") φμ -+ φ as μ —» °°.

We define Ηφ on (W) by the relation

(Ηφ)(χ) = lim [Ηφμ](χ) (5.15) μ,—►<»

The existence of the limit (5.15) follows from the continuity of the operator H on "DîR"). To prove the uniqueness of the limit, assume that there exist two sequences: {ψμ} = I and {ψμ} in X(R") that converge to ψ in DîW). Then, by definition, we have, for all \lc\ = 0,1,2

y\k\(<P ~ Ψμ) -» ° as μ ^ oo

7|*|(<P - Ψκ) -» 0 as v -► oo

Now

7|*|(Η,Ρμ - Ηψρ) = Tltl (Η(<Ρμ. - ψν)) s Cp,Cp2 · · · Cp„ γ|4|(φμ - ψ„)

- Cp[7|t|(<pM -φ) + γ^\(φ - Φμ.)] -+ 0

as μ, ν -» oo V |¿| E N. Therefore the limit (5.15) exists and is unique. In Section 5.5 it will be shown that

limit in the right-hand side of (5.15) is

ax«¡-»o π" ;k—/il>«¡ <p(f)<ft

, , . ,>ο " , = 1.2,3... \[{x¡-t¡)

ι = 1

The linearity of H is obvious. As a matter of fact we can now prove

Theorem 3. Let H be a mapping defined by (5.15). Then for 1 1 (5.16)

or

H - 1 = (-1)"H (5.17)

144 π-DIMENSIONAL HILBERT TRANSFORM

Proof Since (5.16) holds on X(W), it must hold on T>lp(M"),p> 1, by the corollory to Lemma 1 and Theorem 1.

In view of (5.16),

Ηφ = 0 = > φ = 0

So H defined by (5.15) is one to one, and that this is also onto follows from (5.16). Hence from (5.16) it also follows that

H_1 = ( - l )"H

Since H is linear and continuous, so is H~'. This completes the proof of the theorem. □

Theorem 4. Let φ G Tf^iW). Then

D*[H(p] = H[D*<p] V|*| (5.18)

(5.19)

Proof. We will prove the result for the test functions in X(R"), and the result for the test functions in O^iW) can be proved by limiting process and by the fact that the operator

D* : X>L,(Rn) —> T>LP(W), p > 1

is continuous V \k\ G M. Let

* = Ec<*(n</v)e X ( r ) μ=1 \ί=1 /

Then

Now using Theorem 1 of Chapter 3, we have

¿brf Therefore

¿ ["¡(<Ρμ,)] UÍ) = H¡ [φ*>] (*,·) Vi

r n

D*[H<p](*) = Σ ^ Π [Ή*?] W

μ=1 /=1

\μ=1 Ϊ1<® .1=1

M

= Ηϋ*φ V<p G X(R")

This completes the proof of the theorem. D

THE HILBERT TRANSFORM OF A TEST FUNCTION IN X(R*) 145

Theorem 5. 5 Let / C LP(R"), p> l,n> I. Define the operator H* on Lp(Rn) by

(H*/)W = A limn / n.?'" ,, (5.20)

|/,--Λ,-|>ί/>0 (=1,2 n

Then the limit in (5.20) exists in l?(W) sense and almost everywhere sense, and

||H7||„sC„ U/H, (5.21)

where Cp is a constant independent of/.

Proof. Theorem 5 is a very special case of more general result proved in [57]. D

Corollary 2. H*2 = (-1)"/ on Lp(W),p > 1.

Proof. Proof is trivial, since H* = H and

H2 = ( - l ) " / onX(Rn) (5.22)

Now X(R") is dense on l?(W). Therefore in view of the continuity of H, (5.22) is true on lf(R"). Therefore H*2 = (-1)"/ on lf(R"). D

Corollary 3. Let φ be an arbitrary element of T>^(W), p > 1. Then

D * ( H » = H*(D*cp) V | * | £ N (5.23)

Proof. This is true for H by Theorem 4 on D^R") , p > 1. But H = H* on 2)L?(R"). Therefore (5.23) must be true. D

5.2.1. The Hubert Transform of Schwartz Distributions in V^iW), p > 1

If / e DtftW), then the generalized Hubert transform H / of / is defined by the relation

<H/,cp> = </,(-1)ηΗφ> \/φ G 2\,,(ΙΤ), ρ' = - ^ , \<Ρ«*> (5.24)

when Ηφ is defned by (3.17). That is, the Hubert transform H/ of / e T>'¿(W) is a linear functional that assigns the same number to φ £ D^R") as / assigns to (-1)"//φ. It is easy to show from (5.24) that H / defined by (5.24) belongs to Di,(R").

Corollary 4. The operator H : T>^{W) — D'^W) as defined by (5.24) is a linear isomorphism from D[p(R") onto itself and

H 2 / = ( - l ) 7 V / £ D [ , ( R " )

The result follows easily in view of (4.22) and the corollory to Theorem 1.

146 n-DIMENSIONAL HILBERT TRANSFORM

Corollary 5. The operator H : D(,(IR") -► £>(,(W) as defined by (5.26) is a linear homeomorphism with respect to the strong as well as the weak topology introduced on 2>^(R").

Proof. The proof is similar to that given in Chapter 2 for the analogous case n = l .D

Definition. The distribution δ e O¡p(W), p > 1 is defined by

(δ,φ> = φ(0) VLp (W)

= φίΟ,Ο,Ο,.,.,Ο)

Linearity and continuity of δ is evident.

Definition. We now define the Schwartz distribution p.v.^ belonging to "D^iU"), n > 1, as follows:

(ρ.ν.-,φΥ) = lim ί — — — d x V ^ e i V í R " ) . 1<P<°° \ X I max«,—0+ J Χ\Χι· · · Χ„

1=1.2 η

(5.25) If φν -* 0 in 2 V (Rn), then

ρ.\.-,φν(χ) x

= |(H<p„)(0)| -* 0 as v -> °°

In view of the fact that the order of integration in the right-hand side of (5.25) can be switched, we can write

or

p.v

p.v

1 _ X

1 _ X

p.v

p.v

( ' vn:=,

1 t*l-*2 ·

^ *J

•Xn)

Definition. Let / e Tf^W). Then we define the generalized function D*/ T>'lp(W) by the relation

<D*/. φ) = (-l)w</,D*(p> ν φ e Ό ,, p1 = -¡j—, \k\ e N (5.26)

Theorem 6. If/ £ 23|,(Κ"),ρ > l,thenD*(H/) = H(D*/)forall \k\ = 0,1,2

SOME EXAMPLES

Proof. For ψ e T> „-(W) we have

<ΗΟ*/,φ> = <Ι>*/.(-1)"Ηφ>

= (-l)w</,D*(-l)nH<p)

= (-1)Ι*Ι</.(-1)"ΗΕ)*φ>

= (-l)W<H/,D*9)

= <D*H/,«p> D

5.3. SOME EXAMPLES

Example 1. Let us solve the following equation in the space V'^iU").

Hy = Hf + S(xuxi xn)


H2y = H2/ + Ηδ

(- l)"y = ( -1)"/ + Ηδ

Now

<Ηδ,φ> = (δ.(-1)"Ηφ) V* f=1.2 n

= (p-v¿.«p) Therefore

Therefore (5.28) reduces to

(-i)"v = (-i)7 + ¿P.v.(I)

or

Example 2. Let us now consider the solution to the integral equation (operator equation)

y = Hy + f(x), f E T>'LP(W) (5.29)


Hy = H2y + Hf

or

or

If n is odd, then

If n is even, then

Hy = (-])" y+ Hf

y-f = (-\)"y + Hf

f + Hf y = 2

H / = - / If/ given in (5.29) is such that H / = - / is satisfied, then (5.29) is satisfied for all y e O'^W. If, however, / given in (5.29) does not satisfy H / = - / , then there does not exist a solution to (5.29) in T>¡p(Un).

If / given in (5.26) is of the form n n

/ w = n(H*->(*i>-n*''(jCi) 1=1 1=1

where g\,gi,---,gn Ξ Lp and Hi, H2 , . . . , H„ are all one-dimensional Hilbert trans-forms,

WigMxi) = - lim / -^-dti I»,—JT/I>e

then clearly H / = —/ is satisfied. Again, if / is of the form

n n

ηχ) = Υ[(αα,Χχύ + Υ[8.<χύ 1 = 1 1 = 1

then H / = / . Clearly a nonzero function of this form cannot, in general, satisfy the equation H/ = —/. Hence the existence of a nonzero / not satisfying H / = —/ is also demonstrated. For further details of such functions, see [80].

If / e Lf(IR"), p > 1, then the generalized Hilbert transform H / of / is given by

(H/)(*)= Hni ( 1 / ""> / f(f)dt

maxii-.0+ J (Of! - fi)(jt2 - t2) * ■ * (xn ~ tn) \t¡-x,\>t¡ 1=1,2 n

GENERALIZED (n + D-DIMENSIONAL DIRICHLET BOUNDARY-VALUE PROBLEMS 149

5.4. GENERALIZED (n + l)-DIMENSIONAL DIRICHLET BOUNDARY-VALUE PROBLEMS

Let /(JC) be a piecewise continuous and bounded function on the real line R. It is a classical result that the solution to the Dirichlet boundary-value problem

d2u d2u 5? + a? = ° (530)

lim u(x,y) = f(x)

in the upper half-plane Ω = (z = x + iy, y > 0) is

«U,y) = - / j/^ñdt [6, p. 45] (5.31) We extend the notion of Harmonic function to the space R(n+1)+ = {(x,y): x G W, v > 0}. We call u(x, y) a harmonic function in an open region of R(n+1),+, if it is infinitely differentiable at each point of the region and satisfies

n

Uyy(x,y) + ^UXiXi(X,y) = 0 i = l

In this chapter we exploit the above definition of harmonic functions to solve the Dirichlet boundary-value problem in the space R(n+'J'+ with a distributional boundary condition. As it turns out, our solution is quite constructive, and its «-dimensional case is an extension of the corresponding classical Dirichlet boundary-value problem.

Let us consider the Poisson kernel for R(n+1)+, given by

Py(x) = — ^7Tw2. y > 0 (5.32)

where π(η+\)/2

Γ((«+ 1)/2)

The following properties of Py(x) are well known:

JR Py{x)dx = 1 for each v > 0 (5.33)

R"

Let φ e Li(W), q > 1, then Uv(x, y) defined by

t/ ,Cr.y)= / <p(x-t)Py{t)dt

is harmonic in the space Rn+1,+ and \K(x,y)\\L^\W(x)\\L,, y>0 (5.34)

Also ϋφ(χ, y) converges to φ(χ) in the norm of V(U") as y -> 0+ [105, pp. 25-62].

150 «DIMENSIONAL HILBERT TRANSFORM

Theorem 7. Let / E 23£,(Rn), 1 ,>{W), p' = p/ip - 1),

(</(r), Py(x - 0), <P(*)) - if. as y -> 0+

In other words, </(f), Py(x - /)> -► / (0 in V't(Rn) as y -► 0+.

Proof. Using the structure formula for / , we have

l(y) = ((f(t),Py(x-t))Mx)) r

= ] £ ( - l ) | a | ( < / w ( 0 , Z W * - /)>, </>W) (5.35)

where /„ E L?(Rn). Now, using Fubini's theorem and integration by parts, we get

Ky) = £ (fait), f Py(x - t)<pM(x)dx) (5.36)

Let

U9(t,y)= f Py(x-t)<¿a\x)dx

Then it follows from (5.36) that for each fixed y > 0, «φ(ί, y) belongs to C°°(R(n+ υ·+) and that

II^MIL, s lk(e)oL < ". (5-37> ||i/,(/.>) - ^ » W l L - 0 as y - 0+ (5.38)

However,

OfUv(t,y)= f Py(x-t)(p(a+ß\x)dx V|/3| =0,1.2,3,. . .

Therefore

γ,βΙ (tyr.y) - φ(α)(/)) ^ 0 as y - 0+ V|/3| = 0,1,2....

/ Py(x - t)<pw{x)dx -> <p(0,)(f) as y — 0+

JR" From (5.36) and (5.39),

(5.39)

lim /(y) = Y] (/„(f), φ(α)(0> = if. Ψ) V(p E 23 ,

Therefore

y " 1α|=0

(/(O. U - í)> -» / (0 in 2$ (R") as y -» 0+ D

THE HILBERT TRANSFORM OF DISTRIBUTIONS IN D ^ (R"), p > 1 151

Example 3. Consider the Dirichlet boundary-value problem

Π

υ»(χ> y) + J2 υ*Λχ· y) = o. y>0

1=1

lim U(x,y) = f(x) in V'AW), p > 1 (5.40)

Solution. Let us define a function u(x, y) by

U(x,y) = (f(t),Py(x-t)), y>0

Using the structure formula for / , we get

r

U{x,y) = Σ{-\)Μ{Μ),Ό?Ργ{χ - 0) lol=o

r

= Σ(-1)Μυα(χ,γ) M=0

where

Ua(x,y) = {fM(t),D?Py{x - t))

By (5.33), ua(jt, v) is harmonic in R(n+1); therefore u(x,y) is harmonic. In view of Theorem 7, u(x, y) satisfies the distributional boundary condition (5.40).

5.5. THE HILBERT TRANSFORM OF DISTRIBUTIONS IN D^(R"), p > 1, ITS INVERSION AND APPLICATIONS

Pandey and Chaudhary [77] in 1983 developed the theory of Hilbert transform of Schwartz distribution space (Du>)', p > 1, in one dimension using Parseval's types of relations for one-dimensional Hilbert transform. They noted that their theory coincides with the corresponding theory for the Hilbert transform developed by Schwartz [87] through the technique of convolution in one dimension.

The corresponding theory for the Hilbert transform in «-dimension is considerably harder and will be successfully accomplished in this book. As I show here, we can also develop the w-dimensional theory of Hilbert transform to D'(U") by a method analogous to that used by Ehrenpreis [36] to extend the theory of Fourier transform to D'. We can also exploit the result proved in Theorem 8 to give an intrinsic definition of the space H(TH,W)) and its topology. Some applications of these results to solve singular integral equations will be discussed. A related boundary-value problem and its solutions will also be discussed.

5.5.1. The /i-Dimensional Hubert Transform

If / £ L/CM"), p > 1, then it is well known that its Hubert transform (///)(*) defined by

(///)(*) = —n lim max e, - 0+ / π / ( ' ) Α ^ (5.41) π ' J Π/=ι(·*<·-'/)

\l¡-Xj\>ti 1=1,2,3 n

exists a.e. and (Hf)(x) G lf(W). It is also known that there exists a constant Cp > 0 independent of/ satisfying

ll(tf/)Wllp ^ C„||/| |. (5.42)

The existence of the integral in (5.41) and its boundedness property as stated in (5.42) was proved by Riesz and Tichmarsh [99] for n = 1. The results for n > 1 were proved by several authors such as Kokilashvili [57] among others. Riesz and Tichmarsh also obtained the following formula:

(H2f)(x) = -f(x) a.e. (5.43)

for the one-dimensional Hubert transform. In this section I generalize the above inversion formula for n > 1 to the space

l/flR"), p > 1 and then to Schwartz distribution spaces V^U") and O'(W).

5.5.2. Schwartz Testing Functions Space T){Un)

The space TKU"), n > 1 is the Schwartz testing function space of C°° functions defined on W having compact support, and the C°° functions defined on U with compact support will be denoted by T> or D(IR). The topology of D(IR") is the same as defined by Schwartz [87]. Accordingly a sequence {<pm}^=, converges to zero in TKU") if and only if

1 · <P\' Ψΐ' <Ρ3· ■ ■ ■ n a v e m e ' r support contained in a compact set K. 2. φ^\χ) —* 0 as m —♦ °° uniformly for each \k\ = 0,1,2, . . . on arbitrary compact

subset of W.

The space X(W) is defined to be the collection of φ G D(R"), which are finite sums of the form

ψ(Χ) = 5 Z <Ρ»Ί(*')<Pm2(X2) ■ · · ψη,Μη) (5.44)

where <pm¡ G T>, for all /' = 1,2 n. We have the following well-known result:

Lemma 1. The space X(W) is dense in the space WiW), p > 1 with respect to the norm topology of I/flR") [106, p. 71].

THE HILBERT TRANSFORM OF DISTRIBUTIONS IN D^ (R"), p>\ 153

5.5.3. The Inversion Formula

If φ Ε X(R") and φ has the representation (5.44), then n

(//<p)(x) = ^J](//,9m,)U,) (5.45) (=1

where //,(<pm,) = φπν the classical one-dimensional Hilbert transform of <pmi defined by

(#í<Pm,)(*í) = - ^ / -Γ"1 T = <Pm,te)·

We are now ready to prove our inversion theorem.

Theorem 8. Let H be the operator of the classical Hilbert transform as defined by (5.41) in «-dimensions. Then for all / G L^IR"),

( / / 2 / ) to = ( - l ) 7 W a.e. (5.46)

Proof. Equations (5.41) and (5.43) imply that the inversion formula (5.46) is valid for the subspace X(R") of LP(U"). To prove it on LP(W), let us assume that / G I/(Rn) and that {<py}J=, is a sequence in X(U") tending to / in Lp(Rn) as j —> °°. Such a sequence exists by Lemma 1. Then

\\H2f - (-ír/iip = \\H2f - (-ir/ - (H\, - (-ην,οΐι, = \\H\f - φ,) - ( -1)"( / - φ,)||„ (5.47)

Now H : Lp(W) -> Lp(Rn) is a bounded linear operator [57]. Therefore H2 is also a bounded linear operator from L^IR") into itself. By (5.7),

I!//2/ - (- DVII, s Kp\\f - φ;\\ρ - o as j -»co Hence

H2f = (-\)"f (5.48)

in the Lp(Rn) sense, and so a.e. as well. The Testing Function Space T)y(W). A complex-valued function defined on W

belongs to the space T>y>(W), p > 1 if and only if

ψ G C"(R"),

φ(Α:) e L"(R"). V |*| e N,

where

φ<*>(/) = DV(0

= DÍy£---D*"<p(f)

0,,<P=5- '=1.2.3 n

* = (*1.*2 * . )

and n

|*| = ] Γ * „ 4,-eN. i = 1 . 2 n D i = l

5.5.4. The Topology on the Space O^iW)

The topology over Du-OR") is generated by the separating collection of seminorms {?(*)}, 1*1 e N, where

Y(*)(<P) = ( f \<Pik\0\pdt) (5.49)

See [110]. Therefore a sequence φ, converges to φ in Ί\ρ(Η") as j —► oo if and only if

7(t)(V; - φ) -» 0 as y -»oo V |*| ε Ν.

A sequence φ; is said to be a Cauchy sequence in I>ip(UN) if and only if for all

1*1 e N,

7(t)(<Pm - <Pn) -» 0 as m, n -> oo

independently of each other. The space DuiW) (1 < p < °°) is a sequentially complete, locally convex

Hausdorff topological vector space [55]. Note If φ e Ihs(W), then <pw(x) — 0 as \x\ — oo for each |*| £ N [87]. If φ} is a sequence tending to zero in VisiR") as j —> oo, then for each |*| ε f J,

<pf\x) —* 0 uniformly on Rn as j —► oo.

This result is well known [87, p. 200].

Theorem 9. The operator H of «-dimensional Hilbert transform as defined by (5.1) is a homeomorphism from T\p{W) onto itself.

Proof. The result is well known for n = 1; see [99]. We use this fact to prove the result for n > 1. For φ(ί) in Tks(W\ p > 1, let us define

(//,φ)(ί,,ί2,...,/,-,,*,,/,+, /„) (5.50)

<P('l.'2 t¡-\.yi,t¡+l....,tn) -■W x¡ - y, ■dyt

THE HILBERT TRANSFORM OF DISTRIBUTIONS IN D^ (R"), p > 1 155

It is easy to see that if / G EMIR"), then

(Hf)(x) = (//,//2 · ··//,_,//,//,+, · · · / /„/)(*) = (//,(//,//2 · · · #,·-ι///+ι · · · Wn)/)W

(operators Ηι,Η2,Η^,... are commutative). Therefore, for φ G XMR"), p > 1, we have

(Ηφ)(χ) = Ηί(φ(χι,χ2 *,·_,,*,·,*,■+, x„))

where

7p(Xl,X2,...,X¡-i,t¡,X¡+i Xn)

1 \D f rtyi.yz yi-\,t¡,yi+u---.yn)J . , , , Λ«-Ι p / TV¡—7 ^ dy*'''dyi-idy¡+i · · ·dy*

By successive application of Theorem 2 for n = 1, it follows that

φ(χ{,χ2,.... JC/_i,r<,Jti+i *„) G 2*,(R")

When JC], x2,..., x¡- \, x,: +1 , . . . , x„, are kept fixed, then it follows that /9 ft

—(Ηφ)(χ) = Ηί — (φ(χι,Χ2 x¡-l,ti,xi+l,...,x„) (5.51) dx¡ dt¡

= HiHlH2 Hi-lHi+l...HHÍ¡-<p(tl....,tH) dti

By successive application of this result, it can be shown that

D*(//*)(jr) = H(Dk<p)(x) (5.52)

Therefore, using (5.42), we have

||D*(//<P)(*)IIP = | | //φ*φ)(*)| |, =£ C,||D*dl,

Hence

φ G IMR") =>Ηφ& D^iW) (5.53)

In view of the inversion formula (5.6), we have

Ηφ = 0 => y(W),

we have

Η[(Ηφ)(-1)η] = φ (5.55)

Note that ( - 1)"Ηφ G O^iW). Therefore / / " ' exists, and using (5.46), we have

/ / - ' = (-1)"// (5.56)

Since H is linear and continuous, in view of (5.56) H~x is also linear and continu-ous. D

5.5.5. The «-Dimensional Distributional Hubert Transform

For p > 1, assume that / G XMR") and g G XMR"), where - + l = 1. Then it is easy to show that

[ (Hf)(x)g(x)dx = ¡ f(x)(-\)n(Hg)(x)dx (5.57) JR- JR-

In the adjoint notation (5.57) can written

(Hf,g) = (M-\)nHg) (5.58)

We are motivated by the equation (5.57) to define the Hubert transform of distribution in «-dimension.

In conformity with the notation used by Laurent Schwartz, we will denote D[/(R'1), p > 1, or some time abbreviated as T>[j, as the dual space of Thji(W) where

l + i = i P <7

Definition. For / G T>¡j,(W), we define the «-dimensional Hubert transform Hf of / as an element of D ^ R " ) satisfying

(Hf φ) = </,(-\)"Ηφ) V<p G Ι\,(01") (5.59)

Ηφ in (5.59) stands for the classical «-dimensional Hubert transform of <p. It can be easily shown that the functional Hf defined by (5.59) is linear and

continuous on Ί\,(Η").

Example 4. Find H8 where δ G V^W). From the definition (5.59) we have

φ(ί)ώ = δ . ( - ΐ ) '

f " JR" (*! - / ] ) · · -(X„ - /„)>

= l (-Wp [ f^dt

n"(-\)n JK»ht2--tn

THE HILBERT TRANSFORM OF DISTRIBUTIONS IN ©^(R"), p > 1

Therefore

Ho = —p.v. = — p.v. TT" \t\h---t„) ir"y

157

(5.60)

Example 5. Find

H I p.v.

Operating both sides of (5.20) by H, we get HI)

"'»-¿"HID Hence

« p.i/. = (-ιτ)"δ

Since the operators Η\,Ηι Hn as defined in Section 5.5 are commutative, we see that

P'V\t\h---t„) P'V' W i 2 · · · '<· . /

where i'i, /'2 i„ is a permutation of 1 ,2, . . . . n.

5.5.6. Calculus on 2^ (R")

Let / G D ^ R " ) . Then the distributional differentiation of Di^R") is defined as

<θ7.φ> = </,(-DWD*i>> V 1 (5.61) p- 1

Now we prove the following:

Theorem 10. Let / G D^(R"). Then D*/// = WO*/

Proo/.

<D*///,<p) = <// / . (- l)WD*«p),V

= <//D*/,<p)

Hence Theorem 10 is established. D

Example 6. Solve in "D[j,(W) the operator equation

y=Hy+f (5.62)

where /G D¡j,(R"),n> 1.

Solution. Operating both sides of (5.62) by H, and applying the Inversion Theorem 1, and using (5.22), we get

y[l-(-m = / + / / / (5.63)

For the case where n is odd,

(5.63) =» y = Zlí-L (5.64)

For the case where n is even,

(5.63) =>Hf = -f (5.65)

Therefore solution to (5.62) does not exist if

Hf Φ -f (5.66)

If Hf = —/ is satisfied, then there exists infinitely many solutions, and in this case

/ y = — is a solution to (5.62)

If gi are such that they satisfy

Hgi = gi (5.67)

then

y = | + £>g , (5.68) i= l

where c,'s are arbitrary constant, satisfy (5.65). The fact that there exists nonzero solutions to Hy = v (n even) follows easily,

since

y = <p$y$<P2(y2) · · · <Pn(yn)

+ (//i<Pi)0-i)(//2<P2)(y2) ■ · · (Hn<pn){y„),

where φ, e D, satisfies Hy = y when n is even, and

y = (#i<Pi)(yi) · · · (Hn<pn)(y„) - φ,Ο*,) · · · <p„(yn)

THE HILBERT TRANSFORM OF DISTRIBUTIONS IN D ^ R " ) , p > 1 159

satisfies

Hy = -y

There do exists nonzero y's not satisfying

Hy = -y

when n is even. As an example, if we choose

n n

y = Yl{Hifpi){y¡) + Y[q>,iy¡) /= 1 i= 1

where <p, G T> such that y Φ 0, then it does not satisfy Hy = —y when n is even. It is still an open problem to determine the whole class of solutions to

y=Hy+f

when Hf = —/ is satisfied for n even.

5.5.7. The Testing Function Space H(T)(Un))

A complex-valued C°° function φ defined on R" belongs to the space H(THM")) if and only if φ(χ) is the n-dimensional Hubert transform of some ψ(ί) in T)(M"). Hence φ G Η(ΊΧΜ.η)) <&■ there exists ψ(ί) in O(W) such that

ψ(χ) = — P [ ^-dt = Ηφ (5.69)

where the integral is being taken in the Cauchy principal value sense and (JC — t) in (5.69) is interpreted as

n

i = l

The topology of H(T)(U")) is the same as that transported from the space ©(IR") to Z/(D(IR")) by means of the Hubert transform H. Therefore a sequence <p„ in Η(ΊΧΜ")) converges to zero in / / ( I W ) ) if and only if its associated sequence ψη converges to zero in D(Rn), where Ηψη = φη for all n G IU

Theorem 11. Let Η(Ίχη")) and Thj,(M") be the spaces defined as before. Then

i. HCDiW)) C Du-CR") and H(OiW)) is dense in T>LP(W). ii. Convergence of a sequence in H(T>(W)) implies its convergence in T>v(W).

Hence the restriction of any / G T>\j,(W) to H(O(W)) is in //'(©(IR")). Therefore

H'iTKW)) D O¡j,(W)

Proof, (i) Since TKW) is dense in T>y(W) and

H : Dt,(R") - ^ OisiW)

is a homeomorphism, we conclude that HCD(W)) is dense in DD-OR")· See also [57]. (ii) Let tyj -» 0 in Η(ΊΧβ")). Then there exists a sequence ψ, -> 0 in O(W) as

y —♦ oo such that //<p; = $¡. Now using (5.42) and (5.44), we have

Hf\^cPHf%^o as y-oo □ Remark In view of the Inversion Theorem 1,

H : //(XKR")) -> 2 W )

is linear and continuous.

5.5.8. The «-Dimensional Generalized Hilbert Transform

The generalized Hilbert transform Hf of / £ T>'(W) is defined to be an ultradistri-bution Hf G H'(TKW)) such that

(Hf ψ) = (/,(-1)"" V<p G //(©(R")) (5.70)

where Ηφ is the classical Hilbert transform defined by (5.1). If g G H'CD(W)), its Hilbert transform Hg is defined to be a Schwartz distribution by the relation

(Hg,<p) = (g,(-\)"H<p), φ&ΊΧ,η")

Let g = Hf for some / G T>'(U"). Then

<//2/. Ψ) = {Hf ( -1 )"//*>>. V(p G D(R") (5.71)

= ifH\) = </.(-D»

=>//* = ( -1)πΙοηϋ ' ( ϊ?π)

where / is the identity operator.

Definition. The derivation D*g of an ultradistribution g G //'(DilR")) is defined as

{Dkg,<p) = (g,(-nWDk (5.72)

for every φ G //(©flR")).

Theorem 12. Let / G ü'(Rn). Then

(///)<*> = //(/<*>) (5.73)

THE HILBERT TRANSFORM OF DISTRIBUTIONS IN D^ÍR"), p>\ 161

Proof.

(DkHf, φ) = (///, ( -1 )'*'/)* φ> V

= </ ,(- l)w+nD*//(p) (from (5.52))

= <0*/.(-1)"//φ>

= (HDkf,v) D

Example 7. Solve in D'(IRn),

f + //f = δ« We rewrite the equation in the form

4-iy + »n = s(x) = 6(JC,) * efe) * · · · * δ(χ„) dx\

Then y + Hf = A(*i) * 6(*2) * · · · * δ(*„) + Cte.*3 Jc»)

5.5.9. An Intrinsic Definition of the Space H(7XU")) and Its Topology

This section gives an intrinsic definition of the space //(©(R")) and its topology. We now have some lemmas to be used in the sequel.

Lemma 1. Let {<p„} =1 be a sequence of functions tending to zero in Dy(R") as V —>oo;

7(*)(<PiO -» 0 as v -► oo V |fc| G N

Then for each \k\ = 0,1,2

φ^ -» 0 as v ->oo uniformly V* G R"

Proo/. The lemma is well known [87], but a very simple proof can be given as follows:

φ<*>(*) = (δ(ί), <p(k)(x - 0) V 0, an r = (n , r2 r„), and \r\ = r, + r2 + · · · + r„ such that

\<p«\x)\ < C7|'r,((p(i>U - 0) [HO, p. 8-19]

< Cy[r^k\t))

162 »-DIMENSIONAL HILBERT TRANSFORM

where

η'οι = T|o| and yfa = max γ ω \j\^\r\

Therefore

\<PÍk)(x)\ s Cy¡r^\t)) - 0 as i, - » *

independently of JC. D

This proof makes use of the fact that S(x) G 2 \ Ρ ( Κ " ) , which amounts to proving Lemma 1. An independent proof can be given by Schwartz technique as developed in Chapter 7.

Lemma 2. Let tp(t) G TKU"). Then as v — 0+ (i.e., y, -» 0+ for all /' = 1,2,3 n),

i. In 2\,(RB), p > 1,

w-yR. φ(0 3Ί >2

('. - x.)2 + y? «2 - Χ2Ϋ + y22

{tn - xn)2 + yl dt -► φ(χ)

ii. In ^ ( R " ) , p > 1,

(t¡ - x¡) (í, - JC,)2 + yf

iii. In TD\j,(W), m = 1,2,..., n, (p > 1)

¿14Ι^Μ

(5.74)

Λ - /» /" „ ^ ° A (5.75)

i=m+l *- ' -1

—* (Hm ■ ■ · W 3 / / 2 / / 1 ψ)(Χ\, *2 -Xm.-Km+l -*n)

iv. In OisiW), m = 1,2,...,«,

Λ

— / 9(0 Π TT"-m Jw 7-, í=m+l

Λ, (';, - Jt/,)2 + >-2

Wißh ' ' " ^Λ <ΡΧ· ■·■«/,··· Je/2-'' ■ * / „ " - 0

(5.76)

dt

(5.77)

Proof, (i) For the proof see [48, p. 400].

THE HILBERT TRANSFORM OF DISTRIBUTIONS IN D ^ (R"), p > 1 163

(ii) Denoting the left-hand-side expression in (5.77) by ir"F(x), we see that

F«>W = -n f Ά ο Π , , {ti~xi 2dt

By successive application of Fubini's theorem and [99, thm. 101, p. 132], it follows that

\\F«\x)\\ < cnpWk)(x)\\p

where cp is a constant independent of φ and y\, y2 yn-

Since the space X(U") is dense in Τ>ν>(Μη), p > 1, it is easy to show that

||F(A)U) - H<pik\x)\\p - 0 as y,, y2 yn -» 0

A much more general result is proved in [48]. (iii) follows as a result of (i), and (iv) is only an elementary variation of (iii) and

can be proved similarly. D Lemma 3. Let z,G C for j = 1,2,3,..., n, where zj = x¡ + iy¡ and x¡, y¡ G R. For φ(ί) E D(R"), define a function F as a mapping from C" to C by

<P(0 i = l (i, - zi)

-dt

if y¡ Φ 0 for all /' = 1,2,..., n, and

F(zuz2 z¡-ux¡,zi+l z„) = -[F(zuz2 z¡-i,x*,z¡+i z„) (5.78)

+ F(zuz2 z¡-i,x^,z¡+i z„)]

if y i = 0, for f, 1 < / < n. Then limj,_0+f(z) converges uniformly to

Y^diry-tH^Hj, ■ ■ ■ Hj, 0+, in view of Lemma 2 (ii),

F(Z) = y\ [ <p(t)

Rcj-xf + y* dt

Π

TT (';„ - * ;J+

« ■ " _ 1 > Ä

i i ( ' Ä - ^ ) 2 + ^ d/

and the result follows in view of Lemma 2 (iv). Now, for our central problem of defining the space H(THU")) intrinsically, we need the following:

Definition. A holomorphic function ψ(ζ) defined on the complex «-space C belongs to the space Ψ if and only if the following properties hold:

PI. φ(ζ) is holomorphic outside the intervals a¡ £ *, < b,, ¿ = 1,2,3 n (the interval depending upon ψ(ζ)).

P2. ifrw(z) = 0 ( , ,, , ¡—r | ,as|z,- |->°°, ί = 1,2 n, for each fixed k \\ζ\\\ζι\··Λζη\)

satisfying |<t| = 0,1,2,3 P3. (a) For each fixed \k\ = 0 ,1 ,2 ,3 . . . , φ^\ζ) converges uniformly Vjt G

R " a s y - > 0 + . (b) For each fixed \k\ = 0 ,1 ,2 ,3 , . . . , φ^(ζ) converges uniformly V* G

liras y - > 0 _ . P4.

Ψ(Ζΐ.Ζ2 Z¡-y,Xj,Zi+i Z„)

= -[*ll(Zi,Z2,...,Zj-uxf,Zi+i,...,Z„)

+ ψ(ζι,ζ2 z,_1,jc1~,z1+1,...,zn)], ι = 1,2,3 n

where

ψ(ζι,ζ2, •••,z,-_i,jcf ,ζ,·+ι, · · · ,ζΒ) = lim ψ(ζ1,ζ2>· · · ,ζ,·,· · · ,z„) y¡—o-

Theorem 13. A necessary and sufficient condition that a function ψ(ζ) defined on the complex n-space C belongs to the space Ψ is that there exists a φ(ί) £ Ό(Μ") satisfying

ΨΟΟ = / n φ ( ? ) —d t Imz, Φ 0, Vi = 1,2,3,. . . ,η (5.79) JR" Yliti - z¡)

9(0 = ρ.υ. / JH· (t\ - zt) ■ ■ ■ ■ (ti - X¡) •••(tn- Z«)

when Im z¡ = 0 for some /', 1 < / < «.

dt (5.80)

Proof. Necessity: If ι/»(ζ) Ε Ψ ί η view of the properties PI, ψ(ζ) as a function of x e R" is a member of T>u>(W) for a fixed y Φ 0 (i.e., for each component of y e R" nonzero). Now from PI and P2 it follows that if {ym}™=, is an arbitrary sequence in W such that ||ym|| —► 0 as m —* °°, then

||ψ(*»(χ + /ym) - ^(Í)(JC + iy,)\\p -» 0

THE HILBERT TRANSFORM OF DISTRIBUTIONS IN D^R"), p > 1 165

as /,m —> °° independently of each other. Therefore {«//(* + iym)}Z=\ ' s a Cauchy sequence in T>w(R"), p > 1. Since T>y>(U") is sequentially complete, there exists a function ψ+(χ) in T>y>(U") such that

lim ψ(* + iym) = ψ+W in T^iW), p > 1

Since {ym} is an arbitrary sequence in W tending absolutely to zero, it follows that

lim ψ(χ + iy) = ψ+(χ) in I M R " ) (5.81) y->0+

Similar arguments show the existence of a function ψ-(χ) in Ou>(W) satisfying

lim ψ(χ + ¡y) = φ-(χ) in T>y,{U"), p > 1 (5.82) y—0-

and hence is the uniform limit [86] with respect to every x G U". In quite a similar way it can be shown that

Ψ(ζι.ζ2 ζ , - , ,^ , ζ ,+ , ζβ) e ÜLfíR")

for each fixed Zj £ C, 1 < y < « and j Φ i. Therefore

ψ(ζ,, z2 , . . . . z,_,, xh z ,+ , , . . . , z„) (5.83)

= τΙΦ(ζι,Ζ2 *,+ , . . . ,z„) + ψ(ζι,ζ2 *f z„)]

belongs to Du>(R"), p > 1 for fixed y\,y2 y,_,, y,+1 yn Φ 0, where y¡ = Im Zj, 1 £ y < «, y # /'. Since ψ(ζ) is analytic outside the interval [a¡,b¡] on the ,Υ,-axis,

ψ ( ζ , , ζ 2 , · · · , * , + ' , · · · , z „ ) - ψ(ζ],ζ2,- · · . * , " , · ■ ■ ,zn)] = 0

outside [α,, ¿>,] on the X¡ real line, V/' = 1,2 n. Using Cauchy's integral theorem, it can be shown that

1 2m

Γ 1 / — ψ(Ζ|.Ζ2 Uj+ l€j),Zj+i Zn)dtj

J -oo ', Zj

= ψ(ζι, z2 , . . . , Zj + ifj,..., z„), Im Zj > 0

= 0, l m z ; < 0 (5.84)

Letting €j —» 0+ in (5.44), we have

1 2m

= Ψ(ζι,Ζ2 Zj z„), Imz, > 0

= 0, l m z , < 0

(5.85)

(5.86)

(5.87)

Similarly we show that

1 / ·" 1 ΊΓ-. \ — ψ(Ζ\,Ζ2 Zj-t.tj ,Zj+i...,Zn)dtj

Δ7Π J—aa lj Zj

= ψ(ζ\ ,Z2 Zj,..., zn), Im Zj < 0

= 0, Irnz, > 0 Therefore, combining (5.85) and (5.86), we get

1 f" 1 Z—: / [<KZl,Z2 Zj-i,tt,Zj+\--.,Zn) 2m 7_„ tj - Zj i i i

- Ψ(Ζ\,Ζ2 Zj-l,tJ,Zj+l ■■■,Zn)]dtj

= ψ(ζι,ζ2 zn), lmzj¥=0, 1 < y < n

In view of Lemmas 2 and 3 and P4, it follows that

ψ(ζ, ,z2,...,Zj-i,Xj,Zj+i,...,zn) (5.88)

1 fx 1 = -Z—.P I [Ψ(Ζΐ,Ζ2 Zj-l,t+,Zj+l,...,Zn)

¿tn ./_«, tj — Xj

- φ(ζ],ζ2 Zj,,r,zj+l z„)]dtj = p.v. Γ θ(Ζι'Ζ2 *J-**-'J+* ^dtj (5.89)

y-oo Xj tj

where

-2τπ'0(ζ,, z2 tj,...,zn) (5.90)

= Ψ(Ζΐ,Ζ2 t¡ Z„)- <p(Zi,Z2,...,tJ,...,Zn)

Clearly 0(Z|, z 2 , . . . , tj,..., z„) = 0 when t¡ &. [a¡, bj\. Exploiting Lemmas 2, 3, and P4 once again, we can prove that

ψ(ζ,, Z2,. . . , Z¡-1,Xj, Zj+X Z/-1,Xi, Z/+,, . . . , Z„)

=pv r r T\(zi)dtjdtt ' 7-00 7-00 (Xj - tj)(Xf. - t¿)Z

where z^ = (zi,z2 z„) with z, and z¿ replaced by x¡ and x¿, respectively, for a suitable η(ζ\, z2 t¡,...,t¡,... ,zn) vanishing whenever í, £ [α,, b¡] and f; £ [a¡,bi]. Using similar arguments, one can show that there exists φ(χ) G TH.W) with support contained in o, < f, ^ ft, V/' = 1 ,2 , . . . , « , such that

«K*i •*2 χη) = ρ.υ. ί =,r^—-dt (5.91)

THE HILBERT TRANSFORM OF DISTRIBUTIONS IN D^iR"), p > 1 167

Now using (5.87) and repeating the technique of contour integration, and so on, as used in deducing (5.87), it can shown that there exists φ(ί) G D(R") satisfying

ψ(ζ)= f *° dt (5.92)

when Im z¡ Φ 0 Vy, 1 < y < w. It can be seen during the course of derivation that <p's used in (5.91) and (5.92) are the same. This completes the proof of necessity.

Sufficiency: Assume that <p(/) G THU"), and define a function ψ(ζ) and a mapping from C" to C by relation

ψ(2)= / ^ ρ τ ^ -dt (5.93)

when Im z¡ Φ 0 V/, 1 < y < n;

Ψ(') ■y.v. I JR R . (fl - Z t ) · · · ( /y-1 - Zj-i(tj - Xj) ■ · · (fB - Z„)

dr (5.94)

when Im z¡ = 0 for some j , 1 < y < n. The support of <p(f) is contained in a¡ s r, </>,·,/= 1,2,...,«. From (5.93) and

(5.94), it follows quite easily that PI, P2, P3, and P4 hold. D

To demonstrate one-to-one correspondence between the space Ψ and HCD(W)), we can define the space Η(ΊΧΜ")) in genuinely intrinsic way as follows:

Theorem 14. A C°° function ψ(χ) defined on W is said to belong to the space H(T)(W)) if and only if there exists a holomorphic function ψ{ζ) defined on C satisfying PI, P2, P3, and P4. In other words, ψ{χ) G H(T>{W)) if and only if ψ(χ) can be extended uniquely as a holomorphic function satisfying PI, P2, P3, and P4.

The convergence of a sequence {i/»m(jc)}^=1t0 z e r o i " t n e space //(D(R")) can be

defined in an intrinsic way as follows: A sequence {</*«}„= i i n H(WM")) converges to zero in HCD{W)) if and only if

i. The associated functions ψ„(ζ) in accordance with Theorem 1 are analytic outside a closed n-box Π;=ι \.aj>bj\ of R", or else ψη(χ) is analytic outside a fixed closed «-box Π;=ι \.aj-bj]·

ii. ψ„(χ) -* 0 in OisiR") as m -> ».

Proof. Clearly, if {ψ„(χ)}"=ι is a sequence in TKU") tending to zero in 2 W ) as m —> oo and

*.(*) = p.». /" * " ( 0 dt (5.95)

φη(ζ) = ρ.ν.ί - — M L _ ^ A , I m z , # O V / = 1,2 #ι

168 »■DIMENSIONAL HILBERT TRANSFORM

then \¡im(z) is analytic outside the closed intervals a¡ £ x¡ s ¿>;·, _/ = 1,2,...,«, and

ßf<PW<0

Therefore

D Φη,(χ) = p-v. / „ . _ . dr

||D*ifc,W|L < c„||(/#>|L - 0 as m -> «.

Hence conditions i and ii are satisfied. If however, conditions i and ii are assumed, then there exist closed intervals

a, ^ t¡ ¿ bj containing the supports of all

ΨΜ)= I - -J f φη(χ)

Therefore

W(X)\\P *> -^c„WWp - 0 as /n - , oc IT

That is, <pm(x) —► 0 in Dis(W) as m —> oo. By Lemma 1, <pm(jt) —» 0 uniformly V* G R" as wi —» oo. By property i, all <pm(x) have support in a fixed «-box Π>=ιΙα;·^1· Therefore, if ψ„(χ) —> 0 in H(I>(W)) as m —> oo, then the associated sequence {<Pm} =i tends to zero in D(Rn) as m —» oo.

We have proved that <pm -> 0 in D(IRn) as AW -> oo ^φ. ψ„ _» 0 in W(D(R")) as m —► oo. Thus the conditions i and ii together describe intrinsically the convergence of a sequence {i¡fm}Z=i t o z e r o i n H(D(W)) as m -» oo. Π

EXERCISES

1. (a) Prove that if φ e Du^R"), p > 1, then there exists a constant ^ depending upon it = (*i,Jfc2 kn) such that |φ(Α:)(')Ι s ek uniformly for all t G. W.

(b) Prove that <p(*>(0 —» 0 as |f| —»■ oo for each fixed it G W. (c) Prove that the convergence of a sequence {<p„(f)}°° to zero in Ί\ρ(Μ.") implies

the uniform convergence of the sequence {φ(^\ί)}ζ_] to zero on R". (d) Prove that a distribution of compact support on R" is a subset of (üu.(Rn))'.

2. (a) Prove that the Schwartz testing function space S(U") is a proper subspace of the Schwartz testing function space Du>(R"), p > 1. Hint: Prove that the function l / Π ί ^ ι Ο + xf) belongs to Ois(W), p > 1 but not to the space S(R").

(b) Prove that the space V(U") is dense in the space S(R"). (c) Prove that the space 5(R") is dense in Du.(R"), p > 1. (d) Prove that the space (©u>(R"))' C S\W).

EXERCISES 169

3. Solve the following boundary-value problem:

UXX + Uyy + U22 = 0, Z > 0

lim u(x,y,z) = p.v. (—) in (l\,(IR3))'

Hint: The Hubert transform of Trb(t\,t2) = p.v.(j-).

6 FURTHER APPLICATIONS OF THE HILBERT TRANSFORM, THE HILBERT PROBLEM—A DISTRIBUTIONAL APPROACH

6.1. INTRODUCTION

In Chapter 2 we saw that a general Riemann-Hilbert problem can be reduced to finding a regular analytic function F(z) in a given region, which is compliment of the curve C such that

F+(x) - F-(x) = f(x) (6.1)

where F+(x), and F-(x) are defined as limits of F(z) as z —+ x+ and z —> x~, respectively, and x is a point on C. The concept of limits z —► JC+ and z —» JC~ are associated with the given curve C as discussed in Chapter 2. There is another equation,

F+(x) + F-(x) = —XP) f P^- dt (6.2) m Jet - x

that is associated with (6.1). The solution to (6.1) can also be obtained by solving (6.2), and vice versa. In fact (6.1) and (6.2) have common solutions for F(z) under certain set of condition on F(z). Writing g(x) = ¿ ( f ) / c V&dt = i(Hf)(x), we can write the Riemann-Hilbert problem and the associated Riemann-Hilbert problem together as

F + 0 O - F - ( * ) = /(*) (6.3)

F+(x) + F-(x) = «00 (6-4)

170



INTRODUCTION 171

where g(x) = i{Hf)(x). So f(x) = i{Hg)(x), a.e., where

(///)(*) = -(/>) / ^ - r f r 7Γ Jc X-t

More explicitly we have

and

1 f° /(O g(x)=-XP) l±Ldt (6.5)

7Π , /_«, / - X

f(x)=-XP) f ^-dt (6.6)

Any one of the equations (6.3) and (6.4) can be called a Riemann-Hilbert equation, and the Riemann-Hilbert problem consists in solving any one of the equations (6.3) and (6.4) subject to a given set of conditions.

It is assumed that the given function / is such that its Hubert transform exists a.e.; if / G U(R\ p > l.then

1 Γ fit) F(z)= — / H-t-Λ. I m z * 0

277/ , /_«, / - Z

where the limit F+ix) of F(z) is interpreted in the 1/(0?) sense when y —> 0±. When f(t)GO^(U), then

2m \ ' t — z;

is solution to both (6.3) and (6.4). In this case F±{x) are interpreted as

F±ix) = Hm ^-. (fit), ^ \ in D{,(R) y-*±o2m \ t — x - ly I

Also g = iHf and / = iHg, g,f G ϋ^,(Κ). Now the problem arises to find an n-dimensional analogue of equations (6.3) and (6.4). That is, if/ G 1/(Κ") or D^iW), we want to know what will be the corresponding analogues of equations (6.3) and (6.4).

For n = 2 we will show that an analogue of the equation (6.3) is

F+ + ixi,x2) - F+_(jr,,jr2) ~ F- + (xux2) + F--ixux2) = fix\,x2) (6.7)

and that an analogue of (6.4) is

F+ + (xi,x2) + F+Ax\,X2) + F- + ixl,x2) + F— (xux2) = gitx,t2) = i2(Hf) (6.8)

The solution to (6.7) or (6.8) will be

,( i l.7i)=(_L)J(P)££_^I_dM,2

172 FURTHER APPLICATIONS OF THE HILBERT TRANSFORM

iffGLP(U),p> 1;

F+(x) = lim F(x + iy) in O'y,(U) y—O*

life. D^(IR2), then

™·*-(a)'(**■»>· a-,JL-«>) Imzi Φ 0 and Imz2 Φ 0.

When n = 3, an analogue of (6.3) is the equation

F+ + + (x\X2X-¡) — F + + _(JCI, χ-ι,χ-i)

+ F+ — (xiX2X3) -F+- + (xux2,x3)

- F--i. + (xi,x2,x3) + F-+-(xi,x2,X3)

+ F— + (xi,x2,xj,)-F (xux2,Xi)

= f (6.9)

and that of (6.4) is

F+ + + (xi,X2,X3) + F+ + -(XUX2,X3)

+ F+--(xix2,x3) + F+- + (xi,X2,x3)

(xt.x2.x3) +F- + -(xi,x2,x3)

+ F- + + (xl,x2,x3) + F- + + (xltx2,X3)

= g(xux2,X3) = i3Hf (6.10)

Here

Hf - l (P\ ( H'uh.h) . . Hf τ(Ρ) / — -r- -T at) dt2 dt3

T3 7R3 (X\ - h)(x2 - t2)(X3 - /3) I f / e V^U3), then

F(zuZ2, 23) = — — j (f(tUt2,t3),- — — ) (2m)3 \ (i, - zi)(t2 - z2)(h - 23) /

Imz, # 0 , / = 1,2,3. The sign rule for the left-hand-side expression in (6.9) or (6.7) is as follows: The

sign depends upon whether the number of — signs attached in the subscript of F is odd or even. With odd many - signs in the lower subscript of F, we attach negative sign, and with even many — signs in the lower subscript, we attach + signs. But in the equation (6.10) and (6.8) we have all positive signs attached. These results will be discussed in detail in Section 6.S.

INTRODUCTION 173

In brief we can state that an analogue of the Hubert problem,

F+(x) + FAx) = 8(x) (6.11)

when / e LP(W) or O^W), is

2" Y^F„k(Xx,X2 X„) = g(X\,X2 Xn) (6.12) k = \

In this expression the σ* are n-dimensional vectors consisting of plus and minus signs. Singh [90] points out that the «-dimensional analogue (or extension) of the equation (6.11) is

2"

Y^Fat{-\)k(xx,x2 x„) = g(x\,x2 x„) (6.13) 4=1

In my view it is more appropriate to say that (6.12) is an extension or analogue of

F+(x) + F-(x) = g(x)

and (6.13) is an extension or analogue of

F+(x) - F-(x) = Six)

However, our objective in this section is, as in [18], to discuss the solutions to the Hubert problems of the type

« = 1 F + 0 0 + F-(*) = /<*) (6.14)

n = 2 F+ + (xl.x2) + F—(x1,x2)=f{.xux2) (6.15)

n = 3 F+ + + (xux2,Xi) + F (x\,x2,x3) = f(.Xi,x2,xj) (6.16)

which is quite different from the Hubert problems. It is this problem that was discussed in my paper [18], which I have here named a

Hubert problem. It is actually a misnomer to name problems described by (6.15) and (6.16) and [ 18] as Hubert problems, but for our purposes it will suffice to follow this nomenclature.

Let F(s) be a holomorphic function in the region Im z Φ 0 of the n-dimensional complex space C. Assume that

F+(x) = lim F(z) mOlsiU") (6.17)

and that

F-(x) = lim F(z) mT>L,(R") (6.18) y-> 0 -

with

Z = (Zl .Z2 Zn) = (X\ + iyi,X2 + »>2 Xn + »>«)

Here v —> 0+ means that y\ —> 0+,^2 —* 0+,...,yn —♦ 0+ simultaneously, with a similar interpretation for y —► 0-. Im z Φ 0 means Im z,■ Φ 0 for i = 1,2,3,...,«. We will consider the following Hubert problem or Riemann-Hilbert problem: Let / G T)[j,(U"). We wish to find a function F(z) = F(z\, z2(...,z„) holomorphic and regular in the region Im z, Φ 0 for all 1 = 1,2,...,« such that

F+(x) + F-(x) = f(x) (6.19)

where F+(x),F- (x) are as defined in (6.17) and (6.18), respectively. The convergence in (6.17), (6.18), and the equality (6.19) are interpreted in the sense of O\j,(W). We will show that in one dimension this Hubert problem can always be solved while in higher dimensions a number of compatibility conditions must be satisfied by f(x).

6.2. THE HILBERT PROBLEM

Given a function / on the real line satisfying certain prescribed conditions, we wish to find a holomorphic function F(z) in the complex plane such that

F + (* )+ F_ (*) = /(*) (6.20)

where

F+(x) = lim F(z), z = x + iy

and

F-(JC) = lim F(z) (6.21)

The mode of convergence may be suitably chosen. The solution to the problem in the classical sense is discussed in Chapter 2 and in the distributional sense is given in [77]. We will attempt to solve the /i-dimensional Hubert problem for the distribution space TflsiW).

Let F(z) be a function defined on the one-dimensional complex plane which is holomorphic in the upper half-plane Imz > 0 and also in the lower half-plane Im z < 0 satisfying the following conditions:

1. F(z) = o(l), |v| —► °° uniformly for every x £ R . 2· sup,eB,y>s \F(z)\ < As < 00. 3. F+(x) = limz^i+ F(z) = lim^o* F(x + iy) in T>[j,{U). 4. F-(x) = l i m ^ . F(z) = lim^Q- F(x + ' » in T>'DM).

Then we have

F<z) = 7 ^ (F+(0-F-(r), - ! - V Imz Φ 0 (6.22) (2τπ) \ t-z/

THE HILBERT PROBLEM 175

If we consider the convergence in D'(U), then

F& = 7 ^ iF+<'> - F-(')' — ) + PW· Imz φ ° (2τπ) \ t-z/

where F(z) is a polynomial in z. From now on, we will consider the convergence in the space I/(R) only, for p > 1. Writing g = F+ - F - , we have

F(Z) = ^ 6 K < ) , ' - X + *

Then we have

and

2τπ \ δ (f - x)2 + y2

lim F(z) = FAx) = h-Hg + Ug] (6.23) y—>0+ Zf

lim F(z) = F-(x) = h-Hg - Ug] (6.24) >—»0- ¿i

where / is the identity operator. A detailed proof of the identities (6.23) and (6.24) is given in [77]. Adding (6.23) and (6.24), we obtain

F+(x) + F-(x)= ~\ng = f (6.25)

Hence, using the inversion formula (3.4), we deduce

g = iHf

So the required function F(z), holomorphic for Im z Φ 0, is given by

F(z) = - Í - / / / / , - i - \ , Im z # 0 (6.26) 2-7Γ \ t - Z /

We now extend the problem to D[,(IR2). Let / e D^(IR2) and let F(zx,z2) be a function holomorphic in the region Im z\ Φ 0, Imz2 Φ 0 satisfying similar conditions as in the case of one dimension:

1. F{zx,z2) = o(l)as |y, | , |y2 | — °°. 2. s u P | > l M | > 0 |F(z,,z2) < As < oo, δ = ( δ , Λ ) .

\yi\3:B2>0

3. limyi_o*.y2-.ot /r(z,,z2) = F + + (jclfx2), l im^^o.^^o- F(zx,z2) = F t - U , , ^ ) , lim>,1_o-.>.2-.o* F(zuz2) = F_ + (jf,,x2), and Ηπι>ι>Λ_ο- F(zi,z2) = F_-(jti,jc2), in ©^(R2), where

zj = xj + iyj, j = 1,2

Then we have

" Ζ 2 ) = ( ά ) ((F++-F+--F-++F-* C . * > = ( T 3 ) ( <F++ - F+- - F- + + F-)Ul(ti_Zil)(t2_z2)) (6.27)

Writing g = F + + - /*"+_ — F-+ + F—, we have

F(zi,z2)=-^-?(g(t), l

( 2 7 π · ) 2 \ 6 ν / ' ( ί 1 - ζ 1 ) ( ί 2 - ζ 2 )

It was proved in [80] that

Jl (2/):

where I\, I2 are the identity operators:

F++ = 7 ^ 2 < - " i + *Ί)( -"2 + »W*.

hgUi.h) = g{t\,xi),

π Jn(t\-xi)

and

-"■ 7 R ('2 - X2)

Similarly we have

(2Ö2 F — = T Í ^ Í - ^ I - ' Ί ) ( - # 2 - i/2)*

Hence/ = F++ + F-- gives

-^[W. / /2- / . /2 ]g = /

That is,

(// - 7)g = - 2 / (6.28)

where / / = H\H2 and / = l\l2 are the two-dimensional Hubert transform and the identity operators on T>y,(M2), respectively. Using the inversion formula (5.46), we obtain

(/ - H)g = -2Hf (6.29)

Adding (6.28) and (6.29), we deduce that

/ + / / / = 0 (6.30)

THE HILBERT PROBLEM 177

Hence, if / does not satisfy (6.30), the solution of the aforesaid Hubert problem does not exist. In [89] it was shown that there do exist functions satisfying (6.30). So let / satisfy (6.30), and let g\, g2 gm in LP(R) be such that they satisfy

y-Hy = 0 (6.31)

Then we have that m

where c¡ (j = 1 m) are constants, satisfies (6.29). Substituting g for F++ — F+- + F-+ + F-- in (6.27), a class of solutions to the Hubert problem is obtained.

Let us now consider the solution to the Hubert problem in the next higher dimen-sion. Let F(zi,z2,z3), where z¡ = x¡ + ¿y; (_/' = 1,2,3) be a function of Z!,Z2,Z3, which is analytic in the region

{(Z1.Z2.Z3): Imzi Φ 0, Imz2 Φ 0, Imz3 Φ 0}

of C3 and satisfies the following conditions:

1. \F(z\,Z2,z-¡)\ = o(\) as ly iLI^U^I —* °°. the asymptotic order being valid uniformly VjCi,jt2,JC3 G R".

2. l i m ^ o - F(zi,z2,z3) = F±±± inLp(IR). y,->±

3. s u p i ^ i ^ x , |F(zi,Z2,z3)l = As < 00, where δ = (8U82,83).

l>3|2«3>o

Now in view of the results proved in [89], there exists g E LP(U) such that

F(Z,,Z2,Z3)= τ Λ τ U M . (2τπ·)3\0 ' n f=i ( í i -z . - ) ,

Therefore using results in [89], we obtain

so that

F+ + + - (i«-' - 1

F (2/)3

f = F+ + + +F— =

(//, +tf,)Cf/2 + if2X//3 + i/3)

^ 3 (//,//2//3 +H1+H2+ H3)g (20


That is,

- 4/J = (+H +Hi+H2+ H3)g (6.33)

Applying the operation (// + H\ + H2 + H$) to both sides of (6.33), we deduce

- 4 / = (H+Hl+H2+ Hi)f = [H2 - (//, + H2 + H3)2]g

= [-1 - (H2 +H¡+ H¡ + 2// , / /2 + 2// , / /3 + 2H2H3)]g

= [-1 + 3 - 2(//,//2 + H2H2 + //,//3)]g

= [2 + 2//(//, +H2+ H3)]g (6.34)

Applying the operators 2H to both sides of (6.33) and adding the result to (6.34), we obtain

-4 /7 / / - 4i7/(// + //, + H2 + H3)f = 2H2g + lg = 0

or

f + (H+Hi+H2+ H3)f = 0 (6.35)

If the given / satisfies (6.35), then and only then a solution to the Hilbert problem exists. If / satisfies (6.35), then the solution to the Hilbert problem can be obtained by solving for g from (6.33) and substituting in the expression for F(i\, z2, z-¡). As we go to higher and higher dimensions, the problem becomes more and more difficult. We leave this as an open problem.

6.3. THE FOURIER TRANSFORM AND THE HILBERT TRANSFORM

It is known that (?Η/)(ξ) = / sgn(£)Τ/(ξ), V/ £ L2(R), where J and H are the operators of the classical Fourier and the Hilbert transformation, respectively. But such a result is not true in general V/ G LP(R), p > 1, ρ Φ 2, in the classical sense. The object of this section is to show that JHf = /" Π"=1 sgn(£)(J:'/)(£) for all Schwartz generalized functions / belonging to the dual space of Du>(R"), p > 1, with respect to the weak topology of the space 5ό(Κ"). The space 5o(R") is a proper closed subspace of 5(R"), the Schwartz test function space of functions of rapid descent, equipped with the topology induced on it by that of 5(R"). In particular, the result is also true V/ e LP(IR"), p > 1. These results are especially important because they are used in proving the fact that a bounded linear operator T from 1/(0?") into itself, which commutes with both translations as well as dilatations, is a finite linear combination of the Hilbert-type transform and the identity operator.

In the sequel we will use the definition for the Fourier and the Hilbert transforms of functions / and g as

(?ί)(ξ) = I f{t)el,idt (6.36)

THE FOURIER TRANSFORM AND THE HILBERT TRANSFORM 179

and

(Hg)(x)=-P f -^-dt (6.37)

respectively, provided that the integrals (6.36) and (6.37) exist. It is also known that

( J W ) ) ( £ ) = /sgn(£)CF/)(£) (6.38)

V/ e L2(R), where

( J / ) ( f ) = l i m / f{t)e*'dt w^°° J-N

By using Fubini's theorem, we can easily show that

σ(Ηφ))(ξ) = isgn(í)(J<p)(í) (6.39)

V<p £ D(IR), where 2?(IR) is the Schwartz test function space consisting of C°° functions with compact supports. Since ΊΧΜ) is dense in L2(R) and J and H are bounded linear operators from L2(R) into itself the result (6.38) follows from (6.39). But the result (6.38) is not true in general for / e V(U), p > 1, p Φ 2. In this note our object is to extend the result (6.38) for elements of (DipiW))' in the weak distributional sense:

(JHf, φ) = V 9 e So(R")

where So(R") is a test function space of functions of rapid descent, namely those C°°(W) functions that vanish at the origin along with their derivative of all orders and for which

sup 1 2 dt^dt1? ...dtkn"

< 00

for all nonnegative integers rti\, mi,..., m„, k\, k-i k„. Clearly S0(U") C S(W). For x £ W we define sgn(;t) = f["=i sgnU). The

topology of So(R") is the same as that induced on S0(U") by S(W) [110, p. 8]. We can see that the space So(R") is nonempty for a function φ defined by

n

φ(χ) = Π <PÍ(X¡) ( = 1

where

φ,(χύ= Γ,-CA.2)-? when x, ΦΟ 10 when x¡ = 0

belongs to S0(U").

6.4. DEFINITIONS AND PRELIMINARIES

Unless otherwise stated, p > 1 and the testing function space Τ>ν(Μη) consist of complex-valued C°° functions defined on W such that

^■(i dk> &* dk"

dx\ ÖX2 dxn

p \ I/P dx 1 < oo

for all nonnegative integers k\,kj,...,k„. We define k = (£|, ki,..., k„) and \k\ = kt + k2 + ■·· + k„.

The topology over T>y(W) is defined by the sequence of seminorms {ym}^=o [67, pp. 169-170], where

ym{<p) =

η ΐ / ρ

|<|<m

The test function space S(W) consists of C°° functions φ(χ) defined on W such that

sup \tm<pik\t)\ = sup fm, m2 .m, ^ — ^ — j : — /-x < 00 dtk< at\}'" dtk

n

The topology on 5(R") is defined by the sequence of seminorms {ym,k}, where

ΎΜΜ$Ψ) = sup|im<p(i)(0|

Clearly SQ(W) C S(W) C I M R " ) [67, pp. 106-107]. It is a well-known fact that O(Un) is dense in D^iW), and since D(Rn) C

S(W) C Du.(Rn), it follows that S(U") is also dense in DuiU"). If a sequence \φν} in S(R") converges to zero as v —» °o, then it also converges to zero in DLP(R"). AS such

S'(W) D CDisiW))'

Each element of CDu>(W))' can be identified by a unique element of S'iR"). Therefore the Fourier transform Jf of / €= (Du^R"))' can be defined as a functional on SiU") by the relation

< J 7 » = < / . J » VVGS(R") (6.40)

Clearly J / e S'(R") [67, p. 118]. The Hubert transform of / e (£>υ>(Κπ))' is defined as a functional on T>is(W)

by the relation

(Hf,φ) = (f,(-$nH<p) Vcp e EMIR")

Clearly / / / e (©^(R"))' [89, p. 248].

(6.41)

THE ACTION OF THE FOURIER TRANSFORM ON THE HILBERT TRANSFORM 181

6.5. THE ACTION OF THE FOURIER TRANSFORM ON THE HILBERT TRANSFORM, AND VICE VERSA

For / e (Du-íR"))' we can see that Hf e ( IMR") ) ' as defined by (6.41) and that J(Hf) £ S'(W) as defined by (6.40). Consequently

< J W ) . = (Hf, Ιφ) V = </,(-1)"// J V<p E S(R") (6.43)

The right-hand-side expression of (6.42) is meaningful as

J(p<=SC T)IP(W) V<p e S(R")

Step (6.42) is now justified. Step (6.43) is also justified as HJy e Du.(R") Vcp e S(R").

The result

HΤΨ = ( - 0 " J ( Π8Ρ»(*.·)Φ(*)) V V/φ G S0(U")

That is, JHf = i" sgn(x) Jf in the weak topology of SÓ(R"). In the same way it can be shown that

HJf = ( - 0 " J(sgn(x)/)

in the weak topology of (J(S0(W))'. In particular, for / G L/(R") we have

JHf = i" sgn(x)Jf inS¿(R")

HJf = (-i)"J(sgn(x)/) in (J(S0(W))

We have therefore proved the following:

Theorem 1. Let / G ( D L ^ R " ) ) ' and J* and H be the operator of the Fourier and the Hubert transformation. Then

i· JHf = i" sgn(x)Jf in the weak topology of the space S¿(R"). ii. HJf = (-i)"J(sgn(x)f)Vf G CDu>(Rn)y in the weak topology of the space

(JíSoíR")))'·

6.6. CHARACTERIZATION OF THE SPACE JXSo(R"))

Proposition. <p(f) G 7(S0(Ua)) <=» /R tk<p(t)dt = 0, and φ(ί) G S(R") V|/t| > 0.

Proof. If φ(ί) G JCSoiR")), there exists a function ψ G S0W) satisfying

<p(0 = / ψ(χ)«"·' Λ JR"

Therefore

/ tk <p(t)dt = i i ty(x)tkeix'dtdx JR" JR" JR"

= [ [ u(x)tkeix'dxdt JR" JR"

= r* / / i\i(k\xyx'dxdt JR" JR"

= ik i (j ^k\x)eix,dx\eitadt

Hence

/ ( 'φ(ί)Λ = /*(2ir)>(i)(0) V|*| > 0 JR»

The result follows. D

Let ß(l/(Rn)) stand for the space of all bounded linear mappings from LP(W) into itself. We say that the operator T G B(L/(R")) commutes with translations if ταΤ = Ττα for all a G R". We define the operator of translation τα by

Taf(x) = f{x\ -ai,X2-a2 x„ - a„)

The operator T G BiL^R")) is said to commute with dilatations Dm if

TDm = D m r

THE p-NORM OF THE TRUNCATED HILBERT TRANSFORM 183

where

-MP

Dmf(x)=(f[m) "/(^Λ ±)

In the above expression rri\, mi,..., m„ are all > 0. One can also see that

ΙΙΓ./ΙΙ, = 11/11, and IUWII, = U/H, V/eL'(R")

Using the part i of Theorem 1, the following theorem is proved in [46,69].

Theorem 2. Let 1 < p < », and let T £ 5(1/(0?")). Suppose that T commutes both with translations and with dilatations. Then there exist constants a,a\,a2,..., djj, ...,b such that

Γ = α/ + ]Γα,· + //,· + Σ a¡iH¡Hi +-- + bH í = l /',;'= 1

where / is the identity operator on LP(W). Here

1 Γ ■ ■■•h x¡ tn) = -P I

TT J-o mm h xi tn) = -p i / ( ' " ' 2 '; ίη)*

- Xi- t¡

and

Hf(x) - - - ' f m

(XI - /Ote -t2)...(x„ -t„) = (HxH2...Hnf)(x)

6.7. THE p-NORM OF THE TRUNCATED HILBERT TRANSFORM

It is shown that a bounded linear operator T from LP(W) to itself that commutes both with translations and dilatations is a finite linear combination of Hilbert-type transforms. Using this, we show that the p-norm of the Hilbert transform is the same as the p-norm of its truncation to any Lebesgue measurable subset of W with nonzero measure.

Preliminaries: Recall that for a function f(x) defined on the real line, the Hilbert transform (Hf)(x) is given by the Cauchy principal value

(Hf)(x) = -(/>) [ —dt (6.45) w JRX-Í

One of the fundamental results in the subject is that (Hf)(x) exists for almost every x if / e 1/(R), 1 1/(ϊ?) is both continuous and linear, and

that

\\Hf\\p £ CpUfWp for 1 1, may also be

defined as

(///)(*) = —n(P) f fit) dt (6.47)

= lim_L [ /(O ¿,

where ε = */e^ + ε^ + · ■ · + «jj, í = (/i, t2 tn) and dt = dt\dt2 · ■ · dt„. The existence of the singular integral in (6.1.3) and its boundedness property

lltf/llps c;\\f\\p (6.48)

were proved by Kokilashvili [57]. Singh and Pandey [89] extended the «-dimensional Hubert transform to the Schwartz distribution space D'(R") [87] and proved that// is an automorphism on the distribution space ©^(IR'1), p> \ [87]. They also obtained the following inversion formula:

( / / 2 / )W = ( - l ) 7 W almost everywhere (6.49)

for / £ LP(W). The inversion formula (6.49) is a generalization of the corresponding one-dimensional result proved by Riesz; see Titchmarsh [89].

In 1972 Fefferman showed the iterative nature of the double Hubert transform [41 ]. In 1989 Singh and Pandey [89] showed the iterative nature of the n-dimensional Hubert transform over the spaces LP(U") and O[j,(W), p> 1. It was shown that

n

H = Y[H¡ (6.50) 1 = 1

where

(//,/)('. ',-..Xi.ti+, tn) = - Í / ( / l ' ' •••• '<" )d t i

π JR *i - '· The operation //, and //,, i,j= 1,2 n, commute with each other.

In the 1960s Gohberg and Krupnik [46] and O'Neil and Weiss [69] had tried to obtain the best possible value C* (= \\Η\\Ρ) of Cp in (6.30). They gave the following upper and lower bounds for C*:

- » * * * & & > ( $

THE /»-NORM OF THE TRUNCATED HILBERT TRANSFORM 185

where

v(p) c o"i ) ' 1 < p < 2

2 :£ p < oo

and 1/p + \/q = 1. Later Pichorides [82] proved that C* = v(p) for 1 ) [—dt, m JEx - t

xGE (6.51)

where £ is a measurable subset of R. It is obvious that there exists a constant Cpß < °o such that

\\HEf\\P ^ Cp£\\f\\p

for every / e L^IR") and moreover the best constant C*¿ < C*. McLean and Elliott [60] proved that

CE = C* = v{p) for 1 < p < oo, (6.52)

provided that the Lebsegue measure of E is not zero. Here we will extend the result (6.52) to «-dimensions. More precisely, we will

show that for the «-dimensional Hubert transform H defined in (6.47),

c;nE = \\HE\\P = \\m\p = c;n = [κρ)]π (6.53)

for every measurable subset E of IR" with nonzero Lebesgue measure. The n-dimensional truncated Hubert transform HE is defined by

(//*/)(*) = —nP I πη {ω „dt, χ GE (6.54)

In view of (6.50) and the fact that

\\H,\\P = C; = HP), 1 ^ ' s «

it is easy to see that

\\H\]p = C; = [v(p)]n

Now we have proved the latter half of (6.53).

6.8. OPERATORS ON L/(R") THAT COMMUTE WITH TRANSLATIONS AND DILATATIONS

Let a = (a\,a2 a„), and let m = (m\,m2 mn) £ W with m¡ > 0 for each i. We define the translation operator

τα : L"(R") -» L"(R")

and the dilatation operators

Dm,Dm. : L'(R")-♦ L'(R")

by raf(x) = / ( * - a) = /((*, - αλ), (x2 - a2) (x„ - a„)),

Dmf(x)-(f[m) l'f(±A ±)

Dm* fix) = ( Π OTi I /("Ί-«1. ™2*2 WnJCn)

Both Ta and Dm are isometric isomorphisms, since

(Ta)_1 = τ-β , (Dmyl = Z>m.

and

I k / H , = \\f\\p, \\Dmf\\p = | | / | | p for every / £ L"(R")

Let β denote the space of all bounded linear operators from Lp(IRn) into itself. Then T £ ß is said to commute with translations if raT = Ίτα for all a £ IR, and similarly it commutes with dilatations if DmT = TDm for all m £ IR with m¡ > O for 1 s i s n. The following lemma, the proof of which is trivial, characterizes an integral operator commuting with translations or dilatations.

Lemma 1. Let K in ß be an integral operator given by

Kf(x)= f K(x,y)f(y)dy, x £ R"

Then

i. K commutes with translations if and only if K is a difference kernel:

K(x,y) = K(x -y,0) = K(0,y - x)

ii. K commutes with dilatations if and only if AT is a Hardy kernel:

K(mx.my) = \T\m>) *<*·>>

where by mx and my we mean (m\X\, m2x2 m„x„) and (nt\y\, m2y2,..., m„yn), respectively.

OPERATORS ON I/(R") THAT COMMUTE WITH TRANSLATIONS AND DILATATIONS 187

Note that the «-dimensional Hubert transform H commutes with both translations and dilatations, since

H = H[t¡2 ■ ■ -H„

and each H, commutes both with translations and dilatations. Actually H is essentially the only integral operator having this property. To prove this we need the following two lemmas.

Lemma 2. Let T E ß, p > 1 commute with translations. Then there exists a unique bounded complex-valued Borel measurable function σ(ξ) satisfying

( f v ) (ξ) = φ(ξ)σ(ξ)

where σ(ξ) E L^R").

Proof. If T G β, then ταΤ (= Γτ„) Ε β for each a E R". The Schwartz testing functions space IHR") is dense in V(W). Let φ E TKR"), and let gm be sequence of C°° functions with bounded support such that ||gm||p = 1, and gm * φ —» φ as m —* oo, in sup norm as well as in L^R") norm [7, pp. 6-8]. Since φ and gm are of compact supports, gm * φ are also C°° functions with compact supports for all m. Therefore in view of the Riesz representation theorem [84, p. 131], there exists a bounded complex regular Borel measure μ on W such that

[T({gm * <p)(y))}(0) = f (f gMMy - x)dx) άμ(γ)

dx gm{x)dy(y)<p(y - x)

gm(~x)(T(p)(x)dx (by Fubini's theorem)

Hence

(gm * Γ(·))(0): TKW) - C

is a bounded linear functional. The Riesz representations theorem asserts the existence of a regular Borel measure μ„ (depending on gm) bounded on W such that

(gm * Τφ)φ) = / φ(-χ)άμ^(χ). φ E D(R") [84,/». 131].

Hence

(gm*7»(v)= f φ(γ-χ)άμη (6.55) JR-

r~yT{gm * φ)(0) = {gm * τ-,ΤφΧΟ)

-L

= (gm * Ττ-,φΧΟ)

Since |/Am(R")| ^ ||T||, we can select a sequence gm in such a way that

lim (gm * T<p)(y) = (Γφ)(ν) (6.56) m—»°°

in L/(R") norm as well as in sup norm. Hence from (3.2), and by selecting an appropriate subsequence {m,} of {m} and letting m¡ —> °°, we have limm._o° fimj = σ(ξ), a bounded complex-valued measurable function

(/ty) (£) = φ(ξ)σ(ξ)φ £ D(Ä") (6-57)

[5, pp. 132,133]. Π

Corollary 1. For T £ ß commuting with translations, there exists σ £ L^R") such that

T/(f) = *(£)?(£). f £ IR", / £ L'ilR") (6.58)

where ~ denotes the operator of Fourier transform.

Proof. Using the definition of the Fourier transform of/ in L/(IR"), where / is treated as a regular tempered distribution in S'(Rn) [7, pp. 131-132], it follows that

? ( f ) = lim / me+ixidx min^-.» Jix.l<N.

where the above limit is interpreted in the sense of S'(Rn) and x ■ ξ is the inner product of x and ξ in R". Since T>(W) is dense in W(W) the result (6.58) follows from Lemma 2, Bergh and Löfström [7, pp. 132-133], and Stein [94, p. 28]. D

Theorem 3. Let 1 < p < °° and T £ ß. Suppose that T commutes both with translations and with dilatations. Then there exist constant a, a¡, a¡j b such that

n

T =al + Y^diHt + Σ a¡jH¡Hj + ■ ■ ■ + bH (6.59)

•<j

where / is the identity operator on LP(R").

Proof. Let T £ ß, 1 , commuting with translations and dilatations. Then from (6.58) we have

f?(£) = σ(ξ)?(ξ). ξ £ R", / £ L"(R") for some σ £ L^íR"). Since

0«/(í) = I I m ' ?(«iit.«2fe rn„i,)

OPERATORS ON L»(R") THAT COMMUTE WITH TRANSLATIONS AND DILATATIONS 189

and T commutes with dilatations, we have σ(ξ) = σ(τη\ξ\,ηΐ2ξ2,... ,τηηξ„), for { = (&.& £ , ) e (R" ) and in, m„ > 0.

Hence

where

σ(ξ) = a(sgn £, , . . . , sgn £„),

s g n ^ = ( - l , i f¿<0 When n = 2, it is easy to see that

1 <r(£i.&) =

22 [σ(1,1) + σ(1, - 1 ) + σ( -1 ,1) + σ ( - 1 , -1)]

+ [σ(1.1) + σ(1, - 1 ) - σ( -1 .1) - σ ( - 1 . - l)]sgnf,

+ [σ(1,1) - σ(1, - 1 ) + σ( -1 .1) - σ ( - 1 , - l)]sgnfe

+ [σ(1,1) - σ(1, - 1 ) - σ( -1 .1) + σ ( - 1 , - l)]sgnf, sgn&

Generalizing this expression we obtain the «-dimensional case

2" n / 1" \ σ(ξ) = 1 ] Γ σ(/Ί, i , , . . . , ί„) + Σ ί Σ ' ;σ( ' ι- '2. · · ·.'«) I sgn£y

j= l \ ι = 1 / ¿=1

+ Σ ( Σ »vwo'i—'»)) sßn & · s s n & + ;,*=! \ . = 1 /

+ (Σ(Π'>]σ(" ''.)|Π88η&

= a + ^ a y s g n ^ y + ^ α,* sgn £y sgn & + · · · + ¿JJsgngy ;=ι >.*=ι y=i

where iy = +1 or - 1 for j = 1,2,...,«. Since Η/(ξ) = J]"=i sgn£y/(£) and //y/(£) = sgn £,/(£), we have the desired result (6.59); see [74]. D

Remark. The n Riesz transforms /?i, /?2 /?„ are defined as

(*/

with

cn = Γ ( ( " , + . 1 } / 2 ) , fo r / e L'(R"), 1 < p < oo[95, p. 57] Γ((« + l)/2)

^ « + 0 / 2


It is easy to see that, in general, they do not commute with dilatations Dm for m = (m\ mn) £ (R"), mx,..., m„ > 0. Hence none of the R/s can be written in the form (6.59), despite the fact that in the particular case when m = {m\,nt\, ... ,m\) with nt\ > 0, the n Riesz transforms commute with dilatations. But only when n = 1, does the Riesz transform IR commute both with translations and with dilatations so that it can be written in the form (6.59).

For a measurable set E C (Rn), define

XE : L"(R") - L'(R")

by

XE/(X) = [ /W. 'f \ 0 , ot

x G £ otherwise

Since any / G Lp(Rn) can be written as

f = XEf + (I - XE)f

the space LP(W) is the direct sum

LP(R") = U{E) ® l /(R" - E)

(The vector space W is said to be the direct sum of the vectors spaces U and V iff every w G W can be written as w = u + v, u G U, v G V.)

Thus the space l / (£) can be treated as closed subspace of L^R"), and for any bounded linear operator T on l /(Rn), we define the truncated operator

'E — XE'XE

For£C(R")andOT,aGRn,

a + E = {a + x:xGE}

mE = {(m\X\,...,m„x„) : x G E}

tnE = {{jnx\ mx„): x G E) m G R

Then we have the following theorem.

Theorem 4. Let E be any measurable subset of R".

i. If T commutes with translations, then

\\Ta+E\\p = ||Γ£||„. Va G R"

ii. If T commutes with dilatations, then

H7-m£|L = \\TE\\„ V m e R ' . m , mn > 0

OPERATORS ON L"(R") THAT COMMUTE WITH TRANSLATIONS AND DILATATIONS 191

Proof. The proof is similar to the one given by McLean and Elliott [60, thm. 2.2] for the one-dimensional case. D

Let μ be the Lebesgue measure on W. Denote by J&(x) the open box centered atjc:

Π

JB(X) = Π(*< ~ δ"*< + 8i)-x = (*i. ·■·.*..) e R" 1 = 1

δ = (δ, δ„) £ W with each δ, > 0

The density of £ at x is defined by

. . . ,. μ(£ Π Js(x)) dE(x) = hm (6.60)

8-0* μ(/δ(*)) provided that the limit exists. Clearly 0 ¿ dE(x) ^ 1. On the one hand, dE(x) = 0 when x $. E (the closure of £); on the other hand, dE(x) - 1 when x £ £° (the interior of £). The Lebesgue density theorem [20, p. 184] asserts that

dE(x) = 1 for almost every x E E (6.61)

Lemma 3. If J is a bounded box centered at 0 and m > 0, then

lim μ ( / Π mE) = ¿£(0)μ(/)

m—»°°

Proof. Let £ be a measurable subset of U". Then for m > 0, we have

μ(/η£) = μ{(/η*ι mx„) : x = (χχ,... ,x„) E E} = ιημ(Ε) and m(E\ Π £2) = (m£0 Π (mE2), for £i,£2 measurable subset of IR". Suppose that J = (-M.M) X · · · X (-Λ/,Μ) (n factors), and let w = M/8, δ > 0. Then mJs(0) = / , and hence

, , m .. μ(£ n y8(0)) .. μ(^£ η 7) t/£(0) = hm = hm — —

s-o+ μ(/8(0)) m-°° μ( / ) which proves the lemma. Then, following Lemma 3 and Theorem 3 in Chapter 3, we have proofs similar to that of Lemma 2 and Theorem 3 in Chapter 3 of McLean and Elliott [60], so we state them without proof. D

Lemma 4. For 1 0.

iii. limm^„ ||(1 - xmE)f\\P = 0 V/ e I/(R"), m > 0.

Theorem 5. Suppose that dE(0) = 1. If T e ß commutes with dilatations, then

117*11, = ΙΙΙΊΙ,.

Since the «-dimensional Hubert transform H commutes both with translations and with dilatations. Theorems 3, 4, and 5 are true for H. Let £ be a subset of W such that μ(Ε) Φ 0. Then there exists an x ε E such that άΕ(χ) = 1, by (6.61). Hence d_j+£(0) = 1. Therefore

\WE\\P = I I » - , + E I I „ = \\H\\P

Thus we have proved the following theorem:

Theorem 6. If μ{Ε) Φ 0, then \\HE\\P = \\H\\P.

6.9. FUNCTIONS WHOSE FOURIER TRANSFORMS ARE SUPPORTED ON ORTHANTS

We characterize the functions in I/(R") and generalized functions in T>{j,{Un), 1 < p < oo, whose Fourier transforms vanish on one or more orthants of R". It is a fairly difficult problem to characterize the functions in 1/(5?") whose Fourier transform vanishes in some orthants of W. Very little is known concerning this problem except the classical Paley-Wiener theorem in one dimension that characterizes the functions in L2(R) having their Fourier transforms vanish for negative values of the variable [54, p. 175]. Later some results for the space L2(R") were obtained by Stein and Weiss [93, p. 112].

Concerning the Fourier transform of a distribution with compact support, it was shown that the Fourier transform of a distribution / with bounded support is a function F{z) = /(exp(-27riz · x)), which may be continued to all complex numbers z as an entire function of exponential growth. The converse is also true [96, p. 15]. For further references, see [96, 94, 97]. But none of those give the explicit characterization of functions in V(W) and distributions in the Schwartz space D^(R"), whose Fourier transforms are supported on a given number of orthants in R". The aim of the present section is to give a complete answer to the problem for functions in LP(W) and distributions in D^(R"), 1 = TT^-n f f^TV^T, \dt ( 6 6 2 )

(27π)η JR. Π>=ι(0 - zj) where z¡ = x¡ + iyj and y, Φ 0 Vy = 1,2,..., n. For a distribution / E T>{j,(W) the corresponding function F(z) is defined as

F(z> = ττ^-η ( / o · T T W ! — ϊ V yj*° v> ( 6 6 3 )

(2m)" \ Uj^tj - ZJ) /

FUNCTIONS WHOSE FOURIER TRANSFORMS ARE SUPPORTED ON ORTHANTS 193

There are 2" different ways in which y —► 0 depending upon the way the various components y¡ of y tend to either 0+ or 0- . Thus we get 2" different boundary values of F(z) as y —> 0. To denote them, we adopt the following notation:

Let σ* = {σ*(1), σ*(2) σ>(ι)} be a sequence of length n whose elements are + and - for 1 < k < 2". Then 2" orthants of R" are denoted by S„k, 1 < k < 2", where

S„, = jjt G R" | x; > 0 if σ*0") = + and x¡< 0 (6.64)

if a*(y) = —, j = 1,2 n >. For example, when n = 2 the various quadrants of R2 are denoted by S++,S-+,S+_, and S--, where

S+- = {x e R2 | JCI > 0 and x2 < 0}

and so on. Similarly the various limits of F(z) as y —> 0 are denoted by

F„t(x)= lim F(z) (6.65)

where 0σ,(Λ = 0+ if σ*(;) = +; otherwise, it is 0~. For / £ O{j,(Rn) or L/(R"), with the limits taken in respective spaces, we have

proved that

2"

/ = Yji- ΐ Γ ' / ν , in D^(R") (or LP(R")), 1 

= ( / fv if / e L'(R") j V<p ε S(R") (6.68)

where φ is the classical Fourier transform of φ defined as

<p(t)e"'xdt Ψ = / '

The space S(R") is the testing function space of rapid descent [1,8,67, 88,109]. From (6.67) we are able to prove the following Paley-Wiener theorem for T>{j,(W)

(or LAR")) (1 < p < oo):

Theorem7. F o r / E !>[?($*"),

*=1

(. I iff ] T ( - l)"*Fat(€) = 0 for £ ε ( J S„t

in some space S(¡(W).

The space So(R") is a subspace of S(U") that is closed with respect to the multi-plication by the function sgn x. The "sgn" is defined as

n

sgn(jt) = JJsgn(jt,·) (6.69)

In the process we prove the M. Riesz and Titchmarsh inequality and many related classical results for V(W).

As an application of our theory, we characterize the solution space of the following Dirichlet boundary-value problem:

ΔΜ = 0 (6.70)

where

with boundary conditions

lim u = Fat in O[j,{W) (or V(W)), 1 < k < 2" (6.71)

Here Fak, 1 ^ k < 2", are arbitrary elements of T>y,{W). Incidently for fixed F„k (1 < jfc < 2") in O'ysiW) (or L/(R")), the system (6.70) and (6.71) has

™ = ( Σ < - ' > ^ < > · Π - ^ > ) · ■■""""" (6·72» as a unique solution.

6.9.1. The Schwartz Distribution Space O^iW)

A C°° complex-valued function φ(χ) on R" belongs to the space DD.(R") iff d"<p(x) belongs to l/(R") for each |a| = 0,1,2, . . . , where a = (ct\, ai,..., a„), a,'s are

nonnegative integers and \a\ = Σ"=ι a¡. The topology over I)y>(W) is generated by the countable family of separating seminorms [67, 87, 110]

7a (<P) = <p(x)\"dx UP

The space T>u>(W) is a sequentially complete, locally convex, Hausdorff topological linear space.

In conformity with the notation used by Laurent Schwartz [87], we will denote O[r(M"), p > 1, as the dual space of EMR") where - + i = 1. It can be shown [67, p. 173] that for / G D'y,(Rn), there exist measurable functions/„ in Lp(Rn) and ak (=N such that

Ia|<* (6.73)

Let S'(R") denote the space of tempered distributions and S(R") the corresponding testing function space of rapid descent [67]. One can see that S(W) C Du^R") and is dense in I M R " ) [67]. Therefore the restriction of / e D^(R") to S(W) is in S'(R"), and each element of O¡j,(W) can be identified with an element of S'(R") in a one-to-one way. Hence with this kind of identification, Oy,(W) C S'(R"). Therefore the Fourier transform / o f / in T>¡j,(W) can be defined by

</.<*>> = </.£> v<pes(R")

Theorem 8. Let / G L^R"), 1 < p < °°. Define

F(x.y)= I f(t)f[ tj *j 2dt

where x = (xx,x2 x„), y = (yuy2, y i■ Φ 0 (_/' = 1,2,..., n). Then we have

,yn), t = (tx,t2,..

(6.74)

, t„) are in R" and

Π f-itj-XjY + y) dt (6.75)

where

\α\,\β\ =0 ,1 ,2 ,

¿ ? -aa' da* θχψθχψ axS·· ^

a* d*

dypn ft,

and the ay's and /3/s are non-negative integers. Also F(x, y) and d"d& F(x,y) are continuous functions of x,y e W. Thus F(x,y) G C^R2").

196

Proof. Set

FURTHER APPLICATIONS OF THE HILBERT TRANSFORM

u(x,y) h - xi (tj-xj)2 + y2j

dt

/ ( , + χ ) * Π ( ^ ϊ ) Λ

Thus by Holder's inequality we have

yj.

\u(x,y\ s U/H, W'Wh, Hence the integral representing u{x, y) is uniformly convergent V x in IR" and a fixed v £ W having all nonzero components. By using the mean value theorem, we can prove the continuity of F(x, y) and u(x, y) with respect to both x and y. These results are true for arbitrary a. Hence, using a standard classical theorem [100, p. 59], it follows that

* « * » - J ! > * n < ^ ^ dt

Also we have

tfF(x,y)\ a 11/11, W'¡W7, Using the fact that

yj ή + yj

aj + 8J t) + (aj - fiy)2

Vy; e (a¡ - 8j, aj + 8j), we can see that for arbitrary ß the integral representing df F(x, y) is uniformly convergent in an appropriately chosen rectangle lying in the region

{y £ W I \yj\ > 0, ; = 1,2 n} .

Therefore we have

*«">-¿™*n¡¡^ífc3* See [100, p. 59]. Π

Lemma 5. Let *,y,r G R" be such that y¡ Φ OVj = 1,2 « .For / G D^flR") (1 ) = ( n o . Π [(o - *J) I « o - *y)2 + >2)]

d*daxF{x,y) = (f{f\%% Π [(tj - xj) I ((tj - Xjf + yj)\

Proof. Since

(6.76)

(6.77)

ft *' V ■> e L«(R")

as a function of f for a fixed x and y, and / £ O'U(W), the dual of \J(W) ( - + - = 1), F(x,y) is well defined for each x,y G W with y having all nonzero components. Using the structure formula (6.73) for / G T>y,(W), we see that

Í, - Xi

M s * \ J-iVi j) y'

tj - Xi = Σ (AW.HΠ( ΓΛ)Λ/ )· Λei/(R">

Then, using relation (6.72), we obtain

d?%FUy) = Σ (fy(t).$%+yil Ms* \ y=l

(/; - X,-)2 + y2

= (/(ο.^Π \ ;=i L

(',· - xj)1 + y)

{tj-XjY+y)

D

6.9.2. An Approximate Hubert Transform and Its Limit in \J(W)

Some of the results proved in Sections 6.3, 6.4, and 6.5 are proved by Tillmann [97, 98] and Vladimirov [104, ch. 5]. So the results proved in Section 6.3,6.4, and 6.5 are not entirely new. However, our techniques are different in that we make an extended use of the results proved by Riesz and Titchmarsh [99], thereby making our treatment

simpler. Our main results proved in Section 6.6 are new and are not proved anywhere else. In our analysis we heavily rely upon the result that

n

J(Hf) = i" W sgn(*,)(J/) V/ e (Du-dR"))', P > 1

in the weak topology of 50(R"). The space 5o(R") is a subspace of the Schwartz testing function space S(W) such that every element of 5o(K") vanishes at the origin along with all its derivatives. The topology of SoíR") is the same as that induced on So(R") by S(R").

Let H be the operator of the classical Hubert transform from LP(W), p > 1, into itself defined by

(///)(*) = Hm —f f{0 -dt max .,-.<> ΤΓ" J\tj-Xj\>( 11,= it*/ - tj)

= — P [ —, {(f) dt (6.78)

It is a known fact that the limit exists a.e. [57] and that (Hf)(x) G LP(W). Also

\\Hf\\p^Cp\\f\\p [34,57,89,111] (6.79)

where Cp is a constant independent of / [57,93]. Titchmarsh [99] proved that if / G L/(IR), p > 1, then its approximate Hubert

transform

(//,/)(*) = - / l(°'vZ%dt, y * 0 (6.80) * JR if - xY + y2

exists a.e., and x - t

y'"o IT Ju it - x)2 + y2

It is also known that

lim - ί / Λ ^ \ J{t)dt = (///)(*) inL"(R)

WiHyf)ix)\\p^Cp\\f\\p (6.81)

where Cp is a constant independent of/ and v. Stein and Weiss [94, p. 218] proved similar result for l / (R) over the Lebesgue set of / . We extend the above results to «-dimensions.

Definition. The n-dimensional approximate Hubert transform (Hyf)(x) of / €Ξ U(W),p> 1, is defined by

(">/)W=iLπ{η-χ~λή/ωώ· y¡φ°·V; = 1,2,3 η

(6.82)

Theorem 9. The operator Hy as defined by (6.82) is a bounded linear operator from l/(R") into itself.

Proof. We will first prove the result for n = 2. Let / G L^R2). Then we have

Jfpdxdy) =(J dyj dx\f(x.y)\"\

- ( / > / dy\f(x,y)\"

UP (by Fubini's theorem [48])

so that

where

ll/llP = ll/(-,>-)lli.p;2.p = ll/U,-)ll2.p;.„

H/U. 0II2, = ( / j^ ·^ 1 "^)

ΙΙ/(·.?)ΙΙι,= (7l/(*,y)l"<i*)

i.p:\,P

UP

UP

(6.83)

Now

and

Wf(x,y)\\\,pXP = Lp norm of H/Cy)!!,,,, as a function of y

\\f{x,y)\\2,p-\,p = Lp norm of \\f{x, ·)||2,Ρ as a function of x

If one of the expressions in (6.83) exists, the remaining two also exist. Since

1 r 2

(//,/)(*!. *2)= — / /(θΠ 7 7 ^ R 2 j=\

xi - *i (tj - xjY + y) dt

by (6.81) we have

H(f/,/)(jC,.Jt2)ll, = ll(//>/)Ul.^)lll.P;2.P

1//, Xl Λ 2/(--f2) 2 |7R ('2 --Ϊ2)2 +>-2 i.P;2.P

where Cp is a constant independent of/ and y [101]. But

l/„ '2 Λ 2/(··'^ |7R ('2 - -Ϊ2)2 + >|

Cpll/(fl.0ll2,;l 2,P;1,P

p;i.p

^ C p2 | | / | | p

In view of Fubini's theorem [43],

l|tf,/||„sc* U/H, Thus the theorem is proved for n = 2. Using similar techniques and induction on n, it can be shown that for / S W(W),

WHyfWp s C; U/H, D (6.84)

Recall that the space X(W) is defined to be the collection of φ £ D(W) which are finite sums of the form

φ(χ) = Σφ„ι(χι)φ„2(ΛΓ2) -.. <pm„(x„)

where

<pm/Uy) £ TKU), 1 < j < n

The space X(W) is dense in W(W) [106, p. 71].

Theorem 10. For / G LP(Un), define (///)(*) (the Hilbert transform of / ) and (Hyf)(x) (the approximate Hilbert transform of/) as in (6.78) and (6.82), respectively. Then

lim (Hyf)(x) = (Hf)(x) in I/(R") norm yi.yi >«-»o

Proof. Let <pm be a sequence in X(W) converging to / in L^R"). Then

lim \\f(x) - <pm(x)||, = 0 m—>oo

Now

Hyf ~Hf = Hyf -Hy<Pm+ Hy ψπ, ~ H ψη + (//?„, ~ Hf)

So

\\Hyf - Z//II, < | | // ,(/ - cpm)||, + ||//,<pm - //<pm||, + \\H{<pm - / ) | | , < c ; ||/ - <pj|, + \\H,9m - ΜψΛΡ + c; \\Ψη, - /ιι,

It is a simple exercise to show that \\Hy<pm — Ηφη\\ρ —» 0 as y —»0. Letting y —> 0, we deduce

lim | | / / , / - tf/||, < 2C; | | / - <pj|,

Now letting m —► °°, we obtain

lim \\Hyf - /Z/11, = 0 D >—o

Theorem 11. Let / G LP(W), 1 >2,..., yn be nonzero real numbers. Then

( V ) W = i l / ( °n y¡ (tj - Xj)2 + y)

dt (6.85)

which, as a function of x, belongs to LP(W). ϋ· lUy/Hp ^ Cp | |/ | |p, where Cp is a constant independent of / and y.

iii. H/,,/ - / | | p -»Oasy -» 0+; that is, y\,y2,... ,y„ ->0+.

Proof. The proof is very similar to that given for Theorem 9. We can use the fact that for g G LP(R),

(lygXx) = - / yg(0 (/ - x)2 + y2 dt e L/(R), y Φ 0

lim (/,*)(*) = g(jt) in I/(R) [99]

\\Iyg\\^Cp\\g\\p

The result i can also be proved by using [48, p. 400]. It is easy to see that if/ ε LP(U"),

WiyfWp =s c; ||/||p V/ e L'(R") D

Theorem 12. For / e V(W), p > 1, and x, y G R", define

(Tf)(x) =— i f{t) π" JR> Π; Π Λ

Ok - **)2 + >>2 dt 1 1 ( f / - * , · ) 2 + V2

(6.86)

where 0 < w < «. Then Γ is a bounded linear operator from LP(R") into itself,

ΙΙΓ/ΙΙ, s c ; ||/||p (6.87)

and

lim (Γ/)(*) = (//,//2 .. · HJm+,... /„/)(*)

= (//,//2...//m/)W (6.88)

where / I ( / 2 , . . . , /„ are all one-dimensional identity operators and Hi, //2 //„are all one-dimensional Hubert transform operators.

Proof. The proof of (6.87) can be given by using the technique followed in Theorems 2 and 3 in Chapter 4, and then (6.88) can be proved by using (6.87) and the density of X(W) in Ι/(Κ") [106, p. 71]. D

6.93. Complex Hubert Transform

Let / e l/flR"), 1 < p < ° ° , andz = (z,,z2,.. .,z„) e C such that Im z¡ = y¡ Φ 0 V/" = 1,2,...,«. We define the «-dimensional complex Hubert transform (Hf)(z) of / b y

(Hf)(z)=—n f fi,) dt

τ " JR" fj¡ Uj - XjY + y)

Then we have the following:

Theorem 13. For / € lAR"), 1 < /> < °°, its complex Hubert transform (///)(z) as a function of x belongs to L/XR") for a fixed y with all nonzero components. Also

\\Hf\\p < (2Cp)n | |/ | |p, (Titchmarsh and Riesz inequality) (6.90)

and

lim (A/Xz) = I f W - - Uj) 1 /(*) in L'(R") (6.91) yi.yj >-.—o+ l * = * /

where

< / / , / ) « = !#» / / c * * > - ^ + ' * ■ > * ,

and

(6.92)

(/;/)(*) = / , / (* , ,...,Xj-Utj,Xj+1 X„) = f(x) (6.93)

Similarly

lim (///)(z) = (... (//, - tfy) . . . ( / /* + //*). . .)/(x) (6.94) ....»—o+ >»-.o_,...

Proof. The proof can be given by using the technique followed in Theorems 9, 10, and 11. D

Theorem 14. For / e IAR"), P > 1, define

Then

<9aF(x) G L/(R")

Proof. We will prove the result for the simple case where 3a = ■£-, and the general result will follow by induction. Now

dxi F(x\,x2 x, n) * k [('. -*,)2 + y\? * L · f{)%\dj-xj)2 + y)

Therefore

d_ dx\

F(x) ^C n-\ t\-y\ {t\ + y\Y L'<R)

I I / ( Í I , . . . ) IUR.- . ) [48, p. 401]

7Γ c;^\\f\\P □

Corollary. For / G ©^(R"), p > 1, and fixed real numbers y\,y2,---,yn different from zero, define

(iy - *,)2 + y)

Then F(x) E 1/(R").

Proof. Using the structure formula (6.12)

/ = Σ *"/« |o|£*

where each /„ G l/iR"), and Lemma 5, we have

F(x)= Σ#( / . (0 .Π |α|=0 \ ;=1 L

The result now follows in view of Theorems 3 and 9. D

(6.96)

Theorem 15. For / S D ^ R " ) , p > 1, and y¡ Φ 0, 1 < ;' < n, define

(//,/)(*) = FW

as defined in (6.96). Then

lim (//,/)(*) = (// , . . . Hnf){x) = (///)(*) lyl->0

(6.97)

where the limit is interpreted as the weak limit on "D¡j,(W).

Proof. In view of Theorem 14 and its corollary, (Hyf)(x) can be interpreted as a regular distribution on 2\«(5?")· Therefore, for each φ G Ϊ\ ,(Κ"),

/(0.Π (o· - *J)2 + y)

= ( ( Σ */■<'>. Π \ \ |o|=0 j=\ L

= Σ ((/«<'>-#Π |o|=0 \ \ >=1

= ílU (MO. Π |α|=0 \ \ y=l

= Σ((/-ω.Π |α|=0 \ \ ;=1

Mx)

Xj-tj

XJ - lj (tj - xj? + y)

(tj - Xj)2 + y)

(tj - Xjf + y)

¿(-1)|α|//α(0.^(5?φ(Χ))Π y=i L

) .φ(χ)> [67, p. 175]

\ , <p(jc) \ [ Lemma 5]

,(-οχΥφ(χΥ)

dx\ (6.98) (0 - XJY + y)

Since

/„ e V(W) and <?>(JC) G DL»(RB)

by using the duality theorems and the limiting processes, the switch in the order of integration is justified. Now letting \y\ —► 0 in (6.98), we obtain

m

lim ((Hyf)(x), <p(x)) = V(- l ) | a | aa ( i ) , ( - i r / / (<? a 'P (0)> (6.99) W ^ ° | ^ o

The steps in (6.99) can easily be justified in view of Theorem 4. Now using the commutativity of the distributional differentiation da and H [77, 89], we deduce

\\m((Hyf)(x)Mx)) = ( H V <?,"/„(/). <ρ(θ) W^° \ ¿HO /

Therefore

= (Hf,<p)

lim / / , / = / / / in D ^ R " ) D

Corollary. For / e D^íR"), define the complex Hubert transform of/ by

F(z) = - 1 / / ( / ) , ^ j r - i ) , Im z¡ = v; Φ 0 Vy (6.100) ^ \ lly=i(z; -tj)

Then n

lim F(z) = T\(Hj - ilj)f (6.101)

Proof. The proof is similar to the proof of Theorem 15. D

6.9.4. Distributional Representation of Holomorphic Functions

The holomorphic function F(z) given by (6.100) satisfies the uniform asymptotic orders (uniformity with respect to x is assumed here)

|F(2)I=Q((^...U-^) "*'» *-"" Let us now reverse the problem. Let F(z) be holomorphic in y¡ > 0 (j =

1,2 n), that is, on S++...+, and satisfies the relation

sup \F(x + iy)\ <Αδ<°ο (6.102)

and the uniform asymptotic order (with respect to x)

\F(x + iy)\ = o(l), y -» °° (6.103)

Assume also that

lim F(z) = F + + ...+(JC) in D ^ R " ) (6.104) >l.>2 >»—0 +

F+ + - + (.t).rtn , ΓΤ ) = \ „ . . (6.105)

Then by using the technique of [27], it can be shown that

Í F(z), (2iri)n y + + " + v / ' Π"=ι(ζ; - ' ; ) / I 0, elsewhere

where the positive orthant S+ + ...+ = {y & W \ y¡ > 0, j = 1,2,... ,n}. Results similar to (6.105) can be obtained by taking F(z) holomorphic in other of the 2" — 1 orthants and evaluating the corresponding limits of F(z). Let

Ω = {ZELC | I m z ; = ^ # 0 V y = 1,2,...,«} (6.106)

For F(z) ε Ho¿(íl), there are 2" different ways of evaluating l im^o F(z) de-pending upon the various components of y going to either 0+ or 0~. These limits are denoted by F„t(x).

Example 1. When n = 2, there are four quadrants S+ + , S- + , S+-, and S— and four different limits F+ +,F_ +,F+ _ , F - - , where, for example,

S- + = {y e U2 | yt < 0 and y2 > 0}

and

F- + (x) = lim F(z) >,->0-,)>2-»0 +

Let //ο€(Ω) be the space of homorphic functions defined on Ω. Then

M = {F(z) e Ησ€(ίϊ) \ F(z)

satisfies the following conditions (A), (B), &(C)} (6.107)

sup \F(x + iy)\ <AS<«> (A) xteR.\yj\zS>0

|F(JC + iy)\ = o(l) as |>,| |y„| -» « (β)

independently of each other and the asymptotic order is valid uniformly V JC G IR" and

lim F(z) = Fa. (x) in T>UW), k = 1,2,..., 2" (C)

where y —♦ 0σ, means y; —> 0σ,(Λ, 1 < y < n. Then we have the following theorem:

Theorem 16. For any F(z) £ M, we have

where m* = the number of minus signs present in the sequences σ*. For example, when n = 2,

°-¿((F"-f--F-+F--)W·,,,-,,)'^-.)) 6.9.5. Action of the Fourier Transform on the Hubert Transform

I f / eL 2 (R) , then

(Hf)(x) = isgn(jr)/(x) a.e. [94, p. 219] (6.109)

where the Fourier transform J of / is defined by

/ ( * ) = f f(t)exp(tx)dt (6.110) JK

Note that in the right-hand-side expression of (6.109) Stein and Weiss use (—i) in place of ι as their Hubert transform differs from ours by a constant factor only. The result (6.109) can easily be extended to L2(R") as follows:

Recall that the space X(R") consisting of finite linear combinations of functions of the type <p\ (x\ )<p2(Jt2)... <pnUn), where each <p¡(.Xj) £ T)(U), is dense in L2(R") [106, p. 71]. Therefore for/ £ L2(R"), we can find a sequence φη mX{W) such that *l>m(x) —» fix) in L2(R") as w —> oo. Denoting by f the Fourier transform operator, we have

Cf (//(Ψ»)))00 = i" sgnU)(J^m)U) (6.111)

where n

sgn(*) = J | sgn(jc,)

Now letting m —► °° in (6.111) and interpreting the convergence in L2(R"), we deduce

(JHfXx) = i" sgn(x)( Jf)(x) (6.112)

The question now arises whether or not such a result can be proved for the space L/(R"), p > 1. We are able to prove the result (6.112) for p = 2 because of the fact that the Fourier transform maps L2(R") into itself. But such a result is not true in general for p > 1, p Φ 2. If / £ I/iR"), 1 < p < oo, its Fourier transform can be defined, treating / as a regular tempered distribution, as follows:

</. Ψ) = if, Ψ) = [ ίψάχ V<p £ S(R")

where φ(χ) is the classical Fourier transform of <p(t) given by

φ(χ) = / <ρ(0βχρ(/·*)Α

where / ■ x is now the inner product of t and x. For / £ L/(IRn), let φ„ be a sequence in X{W) tending to / in V(W), as m -+ oo. Then we have

lim φ„ = f in S'(W)

Since the Hubert transform H is a bounded linear operator from LP(W) into itself [99], it follows that

lim With,)) = J(Hf) m—»oo

As ψ„ £ L2(R"), from (6.39) we conclude that n

lim /n TT sgnU, )^ = f(H(f)) (6.113) m—»oo A A

7=1

That is,

< W . <P) = Hm <<" sgn(jc)írm(x), φ(χ)) (6.114) m—>oo

= lim i" f sgn(xWm(p(x)dx \/φ £ S(W)

But still we cannot, in general, say that the limit in (6.114) equals /" sgn(jt)/ V/ E LAR"), 1 < p < oo.

We now construct a testing function space So(R") t n a t is a subspace of S(W) closed with respect to multiplication by Π;=ι sSn(*/)· The topology of So(R") is the same as that induced on it by 5(R"). So(R") >s a nonempty subspace of S(R"). All functions in S(W) that vanish at the origin along with all of their derivatives are in 5o(R"). For example,

(11% lo , otherwise

φ(χ) = , , i,= i exp(-*? - x]2). xj*0,Vj=l.2 «,

and

e x P ( ~ l \M )exp(-W2), U I > 1 ,

. 0, U| < 1

are members of SoiR")· The convergence of a sequence to zero in So(Rn) implies its convergence to zero in T>is(W). Therefore the restriction of / G V^W) to S0(R") is in SQ(W). We express this fact by saying that D^W) C S¿(R"). Elements of Vy,(R") cannot be identified with the elements of S¡¡(W) in a one-to-one manner, since S0(W) is not dense in T>y,(Un). Therefore

n

IW) = i" U sgn(Xj)Jf on So(R"), V/ £ L'(R") (6.115)

Because

(J( / / fc) , <p) = (i" sgn(x)(J^X*), <pW> (from 6.112)

= <i"fo,(jr). sgn(x) <pW) Vcp E S0(R")

Now taking the limit m —► °°, we obtain

<7(Hf),<p) = (in/(jc)>sgnU)<p(jc)>

= <i" sgn(jc)/, <p(*)> V<p(jc) E 5o(Rn)

Definition. The Hubert transform Hf of / E D ^ R " ) is defined by

<///, φ) = </,(-1 )"Ηφ) V<p E X\,(R")

where (Ηφ)(χ) is the Hubert transform of φ Ε 2\,(Κ"), given by (6.78).

Definition. The Fourier transform Jf (= / ) of / E D^(R") is defined by

(Ϊ,φ) = (/,Φ) V9ES(Rn) (6.116)

Then we have the following:

Theorem 17. Let / e T>¡j,(U"), 1 = ( J W £ 37« · * ) [from (6.73)]

= / ^ ^ / / / α , φ \ [78] \|o|=Sm /

= Σ { H f a · (—ΐ),β, !ΤΦ> |a|£m

= X](J / / /a , ( -D | a | (w) a <p) [96, p. 9] |a|«m

= 5 3 (Γ sgn(jc)/a(jc), ( - \)Μ(ίχ)αφ(χ)) \a\-<m

= 5 Z <'""/«- ( - l)|a|<?;(J(sgn(f)(p(/)))(jc)> |a|==m

= ( Σ '■"*?/« W- CF(sgn(/)<p(0)W)

= <i"sgn(jc)/(jt)pV(jt)> [from (6.73)] D

Another proof of this theorem is given in [78]. In [78] the result

J W ) ( £ ) = i" sgn(f )(F/) in S'Q(W)

is made use of to prove the fact that a bounded linear operator T from L^R") into itself that commutes with the operators of translation as well as dilatation is a finite linear combination of the identity operator / and the Hubert transform-type operator H\,H7·· ■ ■ ,Hn,H¡Hj,H¡HjHk,...,//.

For / 6 T)[p(W), define a holomorphic function

where yj is the imaginary part of z¡. Then we have the following decomposition theorem:

Theorem 18. For / G T>i,(W), 1 |->0„j(i, >„—0^,,,

Here m* stands for the number of negative signs in the sequence σ*.

Proof. Without loss of generality, we can take n = 2. Then

Now

F+ + (x) = lim F(z) = - \ ((//, - i/, )(//2 - i/2)/) W

[Cor. 5.2] F_ _(x) = lim F(z) = - \ ((//, + //, )(//2 + i/2)/) (JC)

Similarly we have

and

F+.(x) = - \ ((//, - </,)(//2 + tf2)/)(jc) 4

F-+(X) = - i ((//, + ;/,)(//2 - i/2)/) w. 4

Then [F++ - F+- - F-+ + F—](x) = - ^ [ - 4 / ] / ( χ ) = f(x)

and

[F + + + F + _ + F-+ + F—](JC) = i2(Hf)(x).

Using the method of induction, these results can be generalized to any positive integer n. Also

F+ + (x) = - V - i(«,/2 + H2h) - /]/(*) (6.122) 4

where H = H\H2 and / = l\l2. Taking the Fourier transform of equation (6.122), we get

F+ +(f) = -λ- [i2sgn(£,)sgn(&) - <2(sgnf, + sgnf2) - l] / ( f )

= i [sgn(6)sgn(&) + sgn(f,) + sgn(fe) + l] / ( f )

Case 1. f i, f2 > 0:

Case 2. f,, ξ2 < 0:

F+Λξ) = 7[1 - 1 - 1 + 1]}(ξ) = 0 4

£ + + ( f ) = ^[1 + 1 + 1 + U/(f) = / ( f ) 4

Case 3. f, > 0, f2 < 0:

£+ + (£) = 7 [ - 1 - 1 + 1 + Π/( ί ) = 0 4

Case 4. f, < 0, ξ2 > 0:

F + + ( f ) = i [ - l + l - l + l]/(f) = 0

Hence f o r f e S — = {f £ R 2 l i i . f e < 0 }

lo, elsewhere

Thus we have proved the theorem for n = 2. Using the induction, the proof can be given for any n > 1. D

Note that

7 ( f ) , f o r f e S + + U S -lo, J ( f + + + F _ _ ) ( f ) .

elsewhere Likewise

( F + + - F + _ + F - ) ( f ) = | / / ( f ) , forf eS++ US+- US— elsewhere

Similar results hold for all the other possible combinations of F„t. Theorem 13 is analogous to the result proved by Tillmann [97, p. 19] for the space H'(U"). However, our techniques is operator theoretic. In other words, it is based upon properties of the complex Hilbert transform and its limit, whereas the techniques used by Tillmann are essentially an outcome of complex integration in C on appropriately chosen Jordan arcs. The space //(Rn) chosen by Tillman [97] is a subspace of Du^R"). The convergence of a sequence in //(Rn) to zero necessarily implies its convergence to zero in the space Du-íR"), and as such, the restriction of any t £ (Du-R"))' to H(W) is in H'(Ú"). That is, in Tillman [97, p. 19] ( IMR") ) ' C //'(R"). However, the advantage of our space (Dis(W))' is that it is a Fourier as well as Hilbert transformable space. By Theorem 17 we are thus able to prove a Paley-Wiener-type Theorem 19. Some special cases of our representation formulas are also proved by Vladimirov[104,ch. 5].

Analyzing in the same manner yields the following results for / G T)[p(U"):

Lemma 6. For / e T>L,(W), 1 < p < °o, and F(z) defined as in (6.123), we have

for 1 < £. < 2", equality in the sense of S¿(R").

Suppose that one of the summands, say, FσΛζ), for some 1 ^ ko ^ 2", is zero V£ e S % . Then, since/ = Ef=i(_1)mi?<r». equation (6.120) implies that /(£) = 0 V£ £ 5 σ . Conversely, suppose that / = 0 V£ e 5σ. . Then again equation (6.120) gives F^ = 0:

<F<7io,<p) = 0 V ^ e 5 0 ( R " )

So

<F%,$>> = 0, VvESodR")

We can generalize the above argument to obtain the following:

Theorem 19. Paley-Wiener Theorem for ©^(R"). Let / e D^(Rn), 1 S¿(R")

* = 1

ha ¡ff e us. (6,23)

0. elsewhere.

THE DIRICHLET BOUNDARY-VALUE PROBLEM 213

£ (. iff Yji-\)mkFak = 0\/ξ G [jSat in J(S0(Un))'

* = i * = i

That is,

/ ¿ ( - ΐ Γ ^ σ ί , φ \ = 0 V^GSodR")

with support contained in |J¿L, 5σ<.

Remark. Lemma 1 (Chapter 6), Theorems 13 and 14 are also true when we replace T>{j,(W) by LP(U") and F(z) by

f(t) _ dt, Imz; # 0 , 7 = 1 , 2 n (2m)" UJy'U.%^j-'i)

and treating LP(W) as a subspace of £>^(IRn).

6.10. THE DIRICHLET BOUNDARY-VALUE PROBLEM

Let F(z) G M as defined by (6.26). Then, by (6.7), we have

^ - Κ ^ Τ φ - ' ^ ' Π ^ - , ) (6'l24) where

Using a method similar to that used in proving Lemma I in Chapter 3, we see that

AF(z) = 0

So we have proved the next theorem.

Theorem 20. AF(z) = 0 VF(z) G M.

Consider the operator equation

ΔΜ = 0 (6.126)

with the following boundary conditions:

lim u = Fa. in £>^(Rn), 1 < k < 2" (6.127)

Then

«*>-δΐ*(έ<-'^.(0.Πτ^_). W.J-1.2 -(6.128)

is in M (from Theorem 11), and it is also a solution of (6.126) with (6.127) as the boundary condition. The fact that F(z) given by (6.128) is a unique solution in M of (6.126) and (6.127) follows from the representation formula (6.128).

6.11. EIGENVALUES AND EIGENFUNCTIONS OF THE OPERATOR H

Let / £ DlpiW), p > 1, be an eigenfunction of the operator of the generalized Hilbert transformation as defined in Chapters 3 and 4, and let λ be the corresponding eigenvalue. Then

Hf = λ (6.129)

Operating both sides of (6.127) by H, we have

H2f = XHf

( -1 ) " / = \Hf = λ(λ/) (by Case 1)

Therefore

A2/ = (-D7 or

λ 2 = ( - 1 ) η

If n is odd, we have

λ = ±i

and if n is even, we have

λ = ±1

We now show that there do exist eigenfunctions corresponding to the possible eigen-values ±1 , ± J .

Case 1. n is even: (a) λ = 1 and Hf = / ; take / = g + Hg where g E O'u(Rn). Then

H(g + Hg) = Hg + H2g =Hg + ( - l )"g = Hg + g

since n is even. So g + Hg is an eigenfunction corresponding to the eigenvalue λ = 1.

EXERCISES 215

(b) λ = - 1 ; take / = g - Hg. We have

Hf = Hg- H2g = Hg- (-l)"g = Hg-g

= ( - l ) [ g - / / g ] = ( - l ) /

So g — Hg is an eigenfunction corresponding to the eigenvalue λ = — 1. Case 2. n is odd: (a) λ = /'; take / = g - Hg,

H[g - Uig] = Hg- iH2g = Hg - / ( - l)"g = i[g - iHg)

Hf = if

Then g — iHg is an eigenfunction corresponding to the eigenvalue λ = /'. (b) λ = - i ; take / = g + ¡Hg. Then Hf = Hg + iH2g = Hg - ig =

Therefore g + iHg is an eigenfunction corresponding to the eigenvalue λ = —/. It is an open problem to find the class of all eigenvalues and eigenfunctions of the

operator H and to find whether or not an eigenfunction expansion of/// can be done interpreting the convergence of the series appropriately.

EXERCISES

1. (a) Find the distributional Fourier transform of sin/. (b) Find the distributional Hilbert transform of cos t. (c) Find (JH) sin t and {HJ) sin / where J, H stand for Fourier and the Hilbert

transform, respectively, taken in the distributional sense. You must specify the space of generalized functions over which your results are valid.

2. Construct a function φ(ί) ε S defined on the real line such that

Í J -a

tm<p(i)dt = 0 Vm = 0,1,2,3, . . .


1 Γ fit)

in the space (T>¡_p), p > 1. 4. Solve the Hilbert problem

F+{x)-F-(x) = 8(t)

Stating various conditions satisfied by F(z), interpreting convergence (a) in T>[j,, p> l ,and(b)D' .

5. 5. Solve the following Hilbert problem in R2,

F+ + (xux2) - F- + (xxx2) - F+-(xux2) + F--(xitx2) = 8(xux2)

where

<δ(χ,,x2). φ(χι,x2)) = φ(0,0) V<p e 1hs{R2)

6. Generalize the problem given in Exercises 4 and 5, and find its solution.

7 PERIODIC DISTRIBUTIONS, THEIR HILBERT TRANSFORM AND APPLICATIONS

7.1. THE HILBERT TRANSFORM OF PERIODIC DISTRIBUTIONS

Let /(/) be a periodic function with period 2τ defined on the real line R. A new definition of the Hubert transform (Hf)(x) of this function is given by

(///)(jt) = lim -(P) f M-dt (7.1)

provided this limit exists; the integral in (7.1) is taken in the sense of Cauchy principal value. It is shown that the definition (7.1) is equivalent to the conventional definition

(Η/)Μ = ¿ C ) / fix - Ocot ( ^ ) dt (7.2)

where the integral in (7.2) is also taken in the sense of Cauchy principal value. We then use our results to extend the Hubert transform to periodic distributions of period 2T defined on the real line, that is, to Ρ'1τ where P2T is the space of infinitely differentiable functions with period 2τ defined on the real line. An inversion formula over the space PL is proved. Many other related results are proved, and our results are used in solving some singular integral equations in the space PL.

7.1.1. Introduction

Assume that /(z) is analytic in the region Im z > 0, and let u(x) and v(x) be its real and imaginary parts on the real axis. Let us further assume that /(z) = 0(ö), |z| —> °° uniformly V0 ^ Θ s IT.

217




Using the contour integration technique over the semicircle in the upper half-plane, we can get

/(*,o) = l(i») / t!L· o)dt

Thus

Therefore

7Γ /_«, x — t π y_oo x -t

u(x) = -(Hv)(x) (7.3)

v(x) = (Hu)(x) (7.4)

where H is the Hilbert transform defined on the real line. Relations (7.3) and (7.4) are called reciprocity relations. Now let us assume that HR(e'w) and H¡(e'w) are the real and imaginary parts of the z-transform of a causal sequence h(n), n s 0. Then over the unit circle we have

H,(eiw) = ! ; ( />) J" HR{ew) cot ( ^ ) de (7.5)

HR(eiw) = h(o) - ! - ( /») Γ H,{em)col {^γ^) de (7·6>

[71, p. 344]. Motivated by the Hubert reciprocity relation given by (7.3) and (7.4), we say that right-hand-side integral in (7.5) is the Hilbert transform of //«(cft) and equals ///(e""), and that (7.6) is an inversion formula for the Hilbert transfonn. Consequently the Hilbert transform of a periodic function /(f) with period 2π is defined as

2TT W)J" f(t)cat(Zjl)dt (7.7)

provided that this integral exists in the sense of Cauchy principal value. By a simple transformation in (7.7), we can show that

(Hf)(x) = ^(P) J* f(x - t)cot I dt (7.8)

This transform appears in the discussion of the convergence and the divergence of the Fourier series of a periodic function and the associated conjugate series [ 112, pp. 20-22, pp. 1-2; pp. 145-147]. The convergence a.e. of (7.8) implies the convergence of the conjugate Fourier series, and vice versa, provided that/ is of bounded variation. There is, however, another good reason to choose (7.7) or (7.8) as the definition for the Hilbert transfonn of a periodic function with period 2ir. In [65, pp. 67-69] it is shown that if t0 is an arbitrary point on a closed contour L and φ(ί) is a given function

THE HILBERT TRANSFORM OF PERIODIC DISTRIBUTIONS 219

satisfying H condition (i.e., |ψ(ί)) — ιΚ'2)Ι ^ A \t\ — f2|'\ μ > 0, A is independent of t\, ti) and <p(/) is an unknown function satisfying

λί^ = φ{ίο) (7.9) ■m JL t - to

then the only solution of (7.9) in the space of functions satisfying H condition on L is

! / « ! > * = - * < , , (7,0) ™ JL t - t0

Using the transformation / = e'e, t0 = e'e°, we have

dt 1 (Θ - θ0\ ,„ ί' ,„ = » cot . ¿Θ + - d% [65, p. 69] t-t„ 2 V 2 / 2

So using further restrictions on φ and ψ that they are periodic functions with period 2TT, (7.9) and (7.10) are transformed into

2~ 1-ίπφ(θ)οοί(^-^)άθ+^Ι φ(θ)άθ = φ(θ0) (7.11)

ί / * ψ(θ)α* (^γ^2) de+Í¿J" *(θ)άθ = " ^ 0 " ) (712) 2

With an additional restriction on φ and ψ,

ι·2π flit

we have

/ φ(θ)άθ= / t//(0)</0 = O

~ ά / * «<e)cot (^τ^) = ψ(θο) (713)

~Ιπφ(θ)οοΐ(^-^\άθ = φ(θ0) (7.14)

Thus (7.13) is taken as the Hubert transform of a periodic function with period 2π, and (7.14) gives its inversion formula. Now using the definition (7.1) which is similar to the definition of the Hubert transform of a function on the real line, we will arrive at a similar definition of the Hubert transform of periodic functions. Butzer and Nessel [12] have extended the definition (7.13) and its inversion formula to the class L ^ (periodic functions with period 2ir satisfying / * π \f(x)\p dx < », p ^ l), and we will be using many of their results proved in [2]. We will see that the uniqueness theorem in general is not valid for the class of functions L^, τ > 0, and p a l .

We now state without proof a lemma that will be used in the sequel.

Lemma 1. Let τ > 0 and t £ R. Then

π (tTT\ l τ-^ l

k*o

l t - 2kT 2kr

f * 0

π /ΐπ\ l ■r- cot ( — = - + hm 2τ \ 2 τ / f π-·00

" I ^ ί - 2 Α τ k*0

t Φ0

(7.15)

(7.16)

where η Ε Ν+. The equation (7.16) is an immediate consequence of (7.15), and (7.15) is an immediate consequence of the results [12, p. 335]. Note that the series in (7.15) and (7.16) converge uniformly over any compact subset of the real line not containing the set of points 2kr Wk = 0, ± 1, ±2, ±3

We now prove

Theorem 1. Let / £ L£T, p s 1. That is, /(f) defined on the real line is a periodic function of / with period 2τ and f*r \f{t)\p dt < °°.

Then the Hubert transform (Hf)(x) of /(f) as defined by (7.1) is

-(P)f fix-I) cot (^~)dt


2TX

Proof. If the limit in (7.1) exists, then

I f"T f(t) x - t

(2π+1)τ

= lim -(/>) /

= lim -(/») / - ^ - Λ, η £ Ν+ η^°>ττ / _ ( 2 η + 1 ) τ Λ - ί

= lim - ( # » ) / i^—^-dt η^°° π 7-(2«+ΐ)τ '

wL 7-5τ ' / - 3 T ί J-T '

= lim V ) Γ f{x-t) i + ¿ —4 J T L "=-N

i /-T Γ , N / = lim -(/>) / f(x - f) - + V —

(7.17)

+ 2ητ 2ΛΤ )

(7.18)

THE HILBERT TRANSFORM OF PERIODIC DISTRIBUTIONS 221

The equality of (7.17) and (7.18) follows from Lemma 1. The series in (7.18) con-verges absolutely and uniformly to γτ cot(í^) in a compact subset of the real line obtained by deleting the origin and the points ±2τ, ±4τ, ±6τ Therefore

(HfXx) = ¿C) f fix - t) cot (^ ) dt (7.19)

provided that the integral in the right-hand side of (7.19) exists. It is a fact that if / G L£T, p > 1 the integral in (7.19) exists a.e. [12, p. 335]. Using the fact that / is periodic with period 2τ, we can also show that

(///)(*) = 1 ( P ) £ f{t) cot ( ( * ~ τ ' ) 7 Γ ) dt (7.20)

We now prove a theorem that will be used in the sequel. D

Theorem 2. If <p(r) is an infinitely differentiable periodic function with period 2τ, then its Hubert transform (Ηφ)(χ) is a periodic function with period 2τ, and

¿¿WH.*) = (ΗφΜ)(χ) (7.21)

Proof.

F(x) = (Ηφ)(χ) = !■(/>) Γ φ(χ - f)cot [ ^ ] dt

= γτ(Ρ) J MX - ') - φ(Χ + t)] COt ( g ) dt

Define ψ(χ, t) as a function of two variables as follows:

= -2φ°\χ) when t = 0

φ(χ, t) is an infinitely differentiable function with respect to t and x. Therefore

FW=¿/V4cot (£)},*

= 1 />' 2TJ0

T φ ( « ( Λ . _ f ) _ (*)(;t + t ) ,

ί cot —- dt t 2τ

or

F(k\x) = γτ{Ρ) J\<p(k\x -t)~ <p(k)(x + r)]cot ( ^ ) dt

= 2^)/_/>(*-r)cot(£)¿,

The fact that F{x) is a periodic function with period 2τ is obvious. D

7.2. DEFINITIONS AND PRELIMINARIES

7.2.1. Testing Function Space P2r

A function φ(/) defined on the real line is said to belong to the space Ρ2τ iff φ(ί) is an infinitely differentiable function with period 2τ. Let us assume that T> is the Schwartz testing function space consisting of infinitely differentiable functions defined on IR with compact support [1,87]. Any testing function φ in ©generates a unique testing function Θ in Ρ2τ through the expression

oo

0(/> = Σ **t -2ητ) <7·22)

Over any bounded t interval there are only a finite number of nonzero terms in the summation in (7.22) because φ has bounded support. Thus we may differentiate (7.22) term by term to get

oo

0(*>(,) = J2 <Pik\t - «2T), k= 1,2,3,... (7.23)

Convergence in the space Ρ2τ is defined as follows: A sequence {0„(O}"= i is said to converge in P2r to a limit Θ if every 0„ is in Ρ2τ and if for each nonnegative integer k the sequence {θ<£\ί)}ζ=ι converges to θ(*'(/) uniformly for all /. The limit function Θ will also be in Ρ2τ, and as such Ρ2τ is closed under convergence. A function £(f) is said to be unitary if it is an element of Ί) and if there exists a real number τ for which

00

Σ ξ(> - 2«τ) = 1 (7.24)

Many such functions exist. An example of one of them for 2τ = 1 is

L· expf — 1 /x( 1 — x)] dx ξ(0 = 7,1 — - 1 < r < 1. £(0 = O > M ^ l (7.25)

J0 e x p [ - l A ( l - x)]dx

DEFINITIONS AND PRELIMINARIES 223

It is easy to see that Ú0

Σ €(k)(t - "2T) = 0, Λ = 1,2,3,... (7.26) „=-oo

[109, p. 315]. We denote the space of all functions that are unitary with respect to a fixed real

number 2τ by the symbol U2T· Clearly, if Θ is in P2T and ξ is in U2T, then ξθ is in D. Also, if {#„}"= i converges in Ρ2τ to Θ, then {£0„}™=1 converges in T> to ξθ. Every Θ in Ρ2τ can be generated through (7.22) from some φ in T>. This is because

00 00

^ Γ ξ(ΐ - 2ητ)θ(ί - 2ητ) = 0(r) ^ ξ(ί - 2ητ) = 0(f) (7.27) η=-οο η=-οο

7.2.2. The Space Ρ2τ or Periodic Distributions

For / £ Ώ', f is said to be a periodic distribution with period 2τ if

f(t - 2τ) = /

(fit), <p(/)) = </(/ - 2τ), φ(ί)> V<p e ©

For example, / ( i ) = sin(^) is a regular distribution with period 2τ, and g(t) = ΣΓ=-οο δ(/ — 2/iir) is a periodic distribution with period 2τ. One can easily see that g(t - 2π) = Σ Γ - - S(f - 2(n + DTT) = g(i). Also

( 00 \ 00

] Γ δ ( ί -2ητΓ) ,φ \ = ^ φ(2ηττ) (7.28) n=-oo / M=-oo

Clearly (7.28) defines g as a continuous linear functional over T), since only finitely many terms in (7.28) survive as φ is of compact support. So g(t) is a periodic distribution with period 2ττ; that is, g(t) £ PL. Theorem 3. If / is a periodic distribution with period 2τ (i.e., / E PL), then it can be identified as a continuous linear functional on Ρ2τ.

Proof The complex number that / assigns to any Θ in Ρ2τ will be denoted by the dot product / · Θ in order to avoid confusion with the number (/, φ) that / assigns to any φ in 2λ This number / · Θ is defined by

ί-θ = (1ξθ) (7.29)

where ξ is any unitary function in U2T. We first show that the right-hand-side expres-sion in (7.29) is independent of the choice of ξ Ε Ρ2τ. It is a fact that

oo

f(0 = Σ -f(ί)ί(ί ~ 2nr) ( 7 · 3 0 )

The summation in (7.30) has only finitely many nonzero terms over any bounded open t interval. We may write for any φ G T>,

( Σ / (0€( ί - 2ητ),φ(ί)\ = Ιπθ,φ(ί)^ξ(ί - 2#ιτ)\

= </('). <ρ(0>

Now let ξ and k be any two elements in U2T. By using (7.30) and the fact that f(i) = fit + 2ητ), we see that for every Θ in Ρ2τ,

<f.&) = (j2f(t)k(t - 2«T).f(0e(r)\

= Σ(ί{ίΜ-2ητ)ξ(ι),θ) n

= Y^ifit + 2«T)*(0f(f + 2HT). 0(/)> n

= Ο2ηοξ(ι + 2ητ),κοθ(ο\

Let θ], 02 £ />2τ and α, β be any two numbers. We have

/ • (αθ ,+ /302) = </,£(αθ, +ββ2))

= α(/,ξθχ) + β(/,ξθ2)

= α ( / · θ,) + ß(f ■ θ2)

Thus the linearity of/ over /^τ is proved. D

Now we show that / is also continuous over Ρ2τ. If {Θ>Χ= i converges in Ρ2τ to Θ, then {ξθν}™=ι converges in T) to ξθ. Hence, a s v - t »

/ · 0, = </,#„>-</,£θ> = / · β

Thus / is a continuous linear functional over PJT-Thus far we have identified a periodic distribution with period 2τ as a continuous

linear functional over Ρ2τ. Now we will show that continuous linear functional over P2T can also be identified as a periodic distribution with period 2τ in such a way that

/■θ = (/,ξθ) (7.31)

still holds true. This is done as follows: If φ in T> will generate a 0 in Ρ2τ through the expression

00

0(0 = Σ <P(f - 2«τ) (7.32) n = —oo

then, by virtue of the fact that / is a functional on Ρ2τ, we define the number (f, ψ) by the relation

</. = / - e (7-33)

This defines / as a functional on T>. Moreover, since both <p(f) and <p(t + 2τ) generate through (7.31) the same 0(f), we have that

(fit), <p(0> = f(0 ■ 0(0 = (fit), <p(t + 2T)> (7.34)

Theorem 4. If / is a continuous linear functional on Ρ2τ and if 0 and φ are related by (7.32), then (7.33) defines / as a periodic distribution with period 2T.

Proof. We first prove that / defined on V by (7.31) is a linear functional

(/, αφχ + βψ2) = f ■ (α0, + βθ2) = a(f ■ 0,) + ß(f ■ fy)

= α</,<Ρι> + /3</.92>

To show that / defined by (7.32) is continuous on D, we assume that {φμ}"=] converges in Ί) to ψ. Therefore {0„}^=, converges in Ρ2τ to 0 defined by (7.32). Since / is a continuous functional on Ρ2τ, we have

</.?,> = / · β „ - » / · β = </.φ>

From (7.34) it follows that / is periodic with period 2τ. D

We have assumed that the definitions (7.31) and (7.33) are consistent. In fact it is so. For by using (7.32) and the fact that f(t) = f(t + 2m), we have V£ e U2T,

(/- ξθ) = /f(t),ξ(θΣ<Ρ(( - 2nr)\

= £ < / ( ' ) , £(0<P(f-2«T)> n

= 5Z</C + 2ητ),ξ(ί + 2ητ)φ(ί)) n

= ( /( ' ) , ¥>(')£>(' + 2nr)\ = </,

Corollary 1. Two distributions / and g in Ρ'1τ are equal if / · 0 = g · 0 V0 S Ρ2τ. In view of the relations (7.32) and (7.33) we obtain for every φ in D,

</.*>> = f - e = g-e = (g,<p)

so that / = g.

Example 1. Let / be a regular distribution generated by locally integrable periodic function of period 2τ. Then

/ • 0 = Γ TAtMt)dt Ja

since

/ • β = </.£»>= I ηθξ(ΟΘ(ί)ώ J — 00

00 /·α + 2ητ+2τ

ra + 2T

/(f + 2wr)£(f + 2«τ)0(ί + 2ητ)Λ

= Σ / Komm*

= / ηι)θ(ί)Σξ« + 2ητ)ώ= f(t)e(t) dt Ja n Ja

The interchange of the summation and the integration is again justified by virtue of the fact that there are only a finite number of nonzero terms in the summation.

Example 2. Let oo

&2Á0 = Σ 8(t - 2nr), T > 0 n=-oo

Then

Ö2T · 0 = 0(0)

%T(/ - τ) · 0(ί) = 0(τ)

It can be readily checked that ¿>2T(f) G P{7. Therefore

Sir ■ θ = (διΤ,ξθ) =(Σ*(Ι- 2ιιτ). ί(0β(/)\

= Σ ξ(2ητ)θ(2ητ) = 0(0) , £(2 π τ ) = θ<°)

since Σ«=-οο £('"~ 2«T) = 1. Similarly we can show that

&2T(t - a) ■ 0(0 = 0(a)

Example 3. Consider

8<*T'(0= ¿ 8 » > ( r - 2 « T )

Then

since

δ^τ)·β = (-1)*β(*>(0) V 0 £ / > 2 T

δ 2 χ · 0 = Σ < _ 1 > * ^ * ^ ( ί ) β ( 0 ] * — ' flr I/=2«T

= ( -1 )* Σ ( * ) Ö(*-p)(0) ¿ f W(2«T) (7.35)

= (-1)*0W(O)

Note that the right-hand-side summation in (7.35) vanishes forp > 1. So the nonzero term in (7.35) is obtained only for/? = 0.

Theorem 5. The space PL is complete. In other words, the limit / of every sequence {/„}"=, that converges in PL is also in PL.

Proof. For every φ in T> there exists a Θ in Ρ2τ related through (7.32). Therefore, from (7.33), and using the assumption that PL is closed under convergence, we have

</,.<Ρ> = Λ · 0

{/v. Ψ >)v=i is convergent as is {/„ · θ}"=1. Since T>' is weakly complete, there exists / £ T)' satisfying

lirn</„, = </, = / - 0

that is, lim/,, · Θ = f · Θ. Again, each /„ £ PL, therefore / £ PL. Also we have

lim/„ · Θ = f · 0 D

Theorem 6. The sequence {/„}"=, converges in PL to / if and only if it converges in T>' to / and each /„ is in PL.

Proof. In the preceding theorem it has already been shown that the convergence in P^ implies convergence in D'. Conversely assume that for each /„ G Ρ[τ the sequence {/„} converges in T>'. It follows that the limit / must also be in Ρ'1τ.

Then, for ξ in U2T and Θ in PiT

Λ-e = </„,#>-</.£»> = / - e D

7.3. SOME WELL-KNOWN OPERATIONS ON /"2τ

1. Addition of periodic distributions on P'l7:

2. Multiplication of a periodic distribution by a constant a:

(af) ·θ = α(/·θ)

So PjT is a linear space. 3. The shifting property of a periodic distribution:

/(f - T) ■ 0(r) = f(0 ■ 0(0

4. The differentiation of a periodic distribution:

f ( k ) - e = (-\)kf-e(k\ ¿ = 1 ,2 ,3 , . . .

One can also show that

d , , , _ . df dg

5. The multiplication of a distribution / in P^ by a periodic testing function p in Pi*.

(pf)-e = f- (ρθ)

That is, / is periodic with period 2τ.

7.4. THE FUNCTION SPACE Vu AND ITS HILBERT TRANSFORM, p > 1

A measurable function /(f) defined on the real line is said to belong to the space Μ1τ if and only if

1. /(f) is periodic with period 2τ, that is, /(f + 2τ) = /(f) a.e. 2. / :T | / ( f) | 'A<oo.

THE FUNCTION SPACE LjT AND ITS HILBERT TRANSFORM,/» a 1 229

It is a fact that every element / G L£T for p > 1 is Hubert transformable in that (///)(JC) exists a.e. in the sense of (7.2) and that (///)(*) G V satisfying

II///II s Cp U/H, (7.36)

[12, p. 336][9, p. 147]. Since (Hf)(x) is also periodic with period 2τ, it follows that (///)(*) G L£T. Thus

we see that the function space L£T is closed with respect to the operation H defined by (7.1). If {(pv(t)} is a sequence of periodic functions tending to zero in Ρ2τ< then ψ(ν\θ —* o as v —► °° uniformly for a fixed k = 0,1,2,3,

Therefore

H^ll, - 0 a s ^ » Hence

\\(H<pvf% = \\H<p<¿% < C„ ||φ<,% - 0 as v -> «

where

= ( / > ) ! ' * ) iMip= / ι / ω ΐ ' Λ , ρ > ι

It is a fact that if / G L£T, p > 1 and g G L|T, <? > 1 such that ± + i = 1, then ¡:T(Hf)(x)g(x)dx = - f_Tf{x)(Hg)(x)dx [12, p. 339]. In duality notation

(Hf,g) = (f,-Hg) (7.37)

The Hubert transform / / / of / G PJT in analogy with (7.37) can now be defined as

/ / / · β = - / · (//0) (7.38)

Therefore

Hf ■ Θ = </, -ξΗΘ) Ve G Ρ2τ, £ G U2T

Since the space Ρ2τ is closed with respect to the operation H it follows that (7.38) defines a functional Hf over /^τ. The linearity of the functional over Ρ2τ is trivial. We prove the continuity of/// as follows:

Let {0„}"=1 be a sequence —► 0 as v -♦ °° in /^τ· Then it also converges to zero in the space Ζ?υ>[-Τ,τ]. where DD>[-T,T1 consists of infinitely differentiable functions defined on [-τ, τ], which along with all its derivative is W bounded, and p > 1. The topology in the space £>υ>[-τ,τ] is defined by a separating collection of seminorms {γ*}Γ=ο where

%<«) = £wk\tWdt ", p>\

[110, p. 8] [87, p. 201].

Of course convergence in Ρ2τ implies convergence in 2?u>[-T,Tj. Thus, by the relation

<///,β,) = </,-//θ,> (7.39)

and the fact that if the sequence 0„ —» 0 in Du-t-τ,τ]. there exists a constant C and a nonnegative integer i ^ O such that

\<Hf <UI = \{f. ~Ηθν)\ < ε·η(θ„) - 0 as v -»co

[110, p. 16]. / / / as defined by (7.38) is a continuous functional over Ρ2τ. The continuity of the

functional Hf over Ρ2τ can be proved more easily by using the fact that if 0„ —♦ 0 in P2r. Then ξΗθν —♦ 0 in D as v —» °°. The fact that / / / can be identified as a periodic distribution with period 2τ is proved by using the relation

00

(///, φ) = Hf ■ Θ where 0 = ^ φ(ί - 2MT) n=-°o

= <///, <ρ(ί + 2τ)>

7.5. THE INVERSION FORMULA

If / e L£T, p > 1, then using the technique in [12, p. 339], we can show that

H2f=-f+yTff{t)dt (7.40)

Therefore, if / E Ρ'2τ, we have V0 e Ρ2τ,

(H2f, ξθ) = (Hf, -ξΗΘ) = {/.ξ(-Ηγθ)

= (f, ξΗ2θ) = //, -ξ \θ - 1 £ e(t)dt

= ( - / « 2;θ ¿£Η) [12, p. 339]. The constant ¿ / Ι τ 0(f) df e Ρ2τ·

Therefore (7.40) is meaningful. Hence we have derived the following inversion formula:

H2f =~f + {-f)i~

where < - / ( £ ) , ξθ) = <-/ , ξ / I T ¿ di) V0 G Ρ2τ. The uniqueness theorem clearly is not true for the Hubert transform of periodic distributions of period 2τ in the space

THE TESTING FUNCTION SPACE Qlr 231

^2τ· ^ e w ' " n o w finí*a space of distributions Q'2j of period 2τ, where the uniqueness theorem for the operator H holds true.

7.6. THE TESTING FUNCTION SPACE Qlr

An infinitely differentiable periodic function φ(χ) of period 2τ belongs to the space Ö2T >f and only if Γ-τ f^ <** = 0· A sequence {φν}™=, in Q2T converges to zero if and only if φ ^ —► 0 as v —> °° uniformly over the interval [—τ, τ] for each * = 0,1,2,3 The Hilbert transform Hf of / E. β2 τ will be defined as

<Hf, ψ) = (f, -Ηφ) V<p ε 02τ (7.41)

Note now the difference in notation, as β2 τ stands here for the dual spaces of β2τ. Clearly, since JJT φ(χ) dx = 0, we have

" 7 = - / V/ e β2 τ

Note that the testing function space β2 τ is closed with respect to differentiation as well the operator H of the Hilbert transformation defined by (7.1) or (7.2). The space β2 τ is clearly a Frechet space. We now show that the testing function space Q-¿T is closed with respect to the operator H of the Hilbert transformation.

Let

ΦΜ = ¿ C ) f fix - Ocot ( ^ ) dt

where φ G β2τ. By Theorem 2 we can say that ψ is infinitely differentiable. To show that /^ τ ψ(χ) dx = 0, we proceed as follows:

Φ(Χ) = ¿ C ) / M * - 0 - <P(x + Olcot ( ^ ) dt

- ÍJT«*. )H(=)}* where ψ(*, ί) is the function as defined in Theorem 2:

φ(χ)άχ = — / dx / Ψ(ΛΓ, Í)Í cot — df T 2 T 7 _ T y0 2T

= 2τ / / ^ - O ^ f c o t —df = 0

as the switch in the order of integration is justified. So

i Φ(Χ) = γ7 i ty(x,t)dxtcol ( ^ ) dt = 0

Note that f *τ φ(χ ± t)dx = 0. Since the testing function space β2 τ is closed with respect to differentiation and the Hilbert transformation, we define Df, the derivative

of / and Hf on Q'2T as

Φί.φ)

Now

Hence

that is, H ' = —H. By using Theorem 2, one can easily show that

D*/// = H(D"f) v / e ρ^τ

Any periodic distribution / with period of 2τ can be identified as a continuous linear functional over Q2T by the relation

/·</> = </■&>>. ξΕΡ2τ,φΕζ>2τ

Therefore the Hubert transform Hf of the periodic distribution / , when identified as a continuous linear functional over 02τ. can be defined by the relation. By using the technique of Section 7.7.2, it can also be shown that any element of Q2r can be identified as a distribution. Now

Hf ■ Ψ = / · (-Ηφ) V) V<p e Ö2T

= </, - / V ) v<p e Ö2T

= < / / / »

-(fi-ΗΫφ) - </.Η2φ)

-if.-ψ)

Hlf = - /

LOCALLY INTEGRABLE AND PERIODIC FUNCTION OF PERIOD 2τ 233

for all period distributions / when identified as a continuous linear functional over 02τ·

7.7. THE HILBERT TRANSFORM OF LOCALLY INTEGRABLE AND PERIODIC FUNCTION OF PERIOD 2τ

Example 4. Let f{t) be a locally integrable and periodic function of period 2τ defined on the real line. Let / be the distribution represented by this locally integrable function. Then by definition the Hubert transform Hf of the distribution / is defined by

(Η/,ξφ) = (/,-ξΗφ) \/φ(ΞΡ2τ

= (m.-tW^inJ Ψ{Χ - t) cot ( ^ ) dt

= / / ( * ) . -ξ(χ) 'ί'Γ'ν^'Ή-ο*) Define the function ψ(χ, t) as in Theorem 2. Now ψ(χ, t) is an infinitely differentiable function with respect to x as well as t, and ψ(χ, t) is also a continuous function of x and /. Therefore

(Η/,ξφ) = - λ f [ /(χ)ψ(χ,ί)ίοο1^ώάχ 2τ./_,../„ ' 2T

The above iterated integral is absolutely integrable. Switching the order of integration, we get

(Η/,ξφ)= ~ f f /(Χ)φ(χ,ΐ)ί COt— dxdt

_ -\_ ~ ~2τ / (P) Í f(x)M* - t) - <p(x + t)] cot ( ^ ) dxdt

tir f{x) dx (/>) / [<p(x - t) - <p(x + t)] cot — dt

~ Γ f{x)(P) f φ(ί) cot{X ~ τθ 7 Γ dt dx

Thus

Hf = ¿( I») f /(x)cot ^ ^ dx

= ¿CP>£/(f-*)«*(£)* Example 5. Find the Hubert transform of the periodic function sin t that is of period 2π. Here 2τ = 2π so τ = ir.

/■"' x - í //(siní) = —-(f) / sin tcot-~-—dt

1 /""

- > f sin(x - t) cot - di

sin x eos / — eos x sin t) cot - </f 1 Γ = — (/>)/ (si

1 Γ · ' v = — —- eos JC / sin / cot - dt

2ττ J0 2 1 Γπ t

= —— cos x / 2cos2-dí 2-tr J0 2

í Jo

— — eos x I (eos / + 1) dt

= — eos x / /sinx = — cosx (7.42)

We can similarly show that

H cos x = sin x

To find the Hubert transform of cos x, we can use the inversion formula

H sinx = — cosx

Therefore

/ / 2sinx = —//COSJC

Using the inversion formula, we have

1 Γ — sinx + —- / sinxdx = — //(cosx) 2W J-TT

Therefore

//(cosx) - sinx, (7.43)

LOCALLY INTEGRABLE AND PERIODIC FUNCTION OF PERIOD 2τ 235

Note that the inversion formulas (7.42) and (7.43) can also be derived by evaluating the integral lim/v-,« ^ f_N £η dt using contour integration technique and then equating the real and imaginary parts.

Example 6. Solve in Q'2ir the singular integral equation

y+Hy = sint (7.44)

Operating the operator H on both sides of (7.44), we get

Hy - v = H sin t

Hy - y = -cost (7.45)

Eliminating Hy from (7.44) and (7.45), we get

2y = sin t + cos t

or

y = -(sin/ + cosí)·

Example 7. Solve

y + / / / = sin t (7.46)

Operating both sides of (7.46) by H, we get

Hy-y'= -cost (7.47)

Differentiating both sides of (7.47), we get

Hy'-y" = sinf (7.48)

Eliminating Hy from (7.46) and (7.48), we get

y + y" = 0

y — A sin t + B cos /

Now we select A and B in such a way that (7.44) is satisfied:

A sin t + B cos t + H(A cos t — B sin t) = sin t

A sin t + B cos t + (A sin t + B cos i) = sin t

A sin t + B cos í = j sin í

So B = 0, A = | , and the required solution is .y = \ sin f.


Example 8. Solve the following integral equation in the space Q'2:

»/■ — (?) The above equation can be written as

JtHf = smY or

So

/ / / = - - cos - ^ + C ir 2

/-mKiv)-^)«?-^/.^?*] — 1 , ^ fl [ XTT ίπ . χπ . /7τ"ΐ ιπ , = - — ( / > ) / |cos — cos y + sin — sin — J cot y Λ

1 /"' . XTT tTT , 4 . /"77*\ =^y_1

s,nTcosydí=^sin(T)

7.8. APPROXIMATE HILBERT TRANSFORM OF PERIODIC DISTRIBUTIONS

Let ΡιΊ be the space of infinitely differentiable periodic functions with period 2τ, and let PL be the space of periodic distributions with period 2τ when the topology over the space Ρ2τ is generated by the sequence of seminorms γ*(0) = sup, |0(t)(/)|, k = 0,1,2,3, It is well known that every / €= PL is identifiable as a continuous and linear functional over Ρ2τ-

We define the approximate Hubert transform

by

Where

F{x,y) of fGPL

F(x,y) = f(t).q(x-t,y), γΦΟ

= {f(t)^(t)q(x-t,y)), y*0

t. Λ l i· V* ( f " 2 n r ) q(t,y) = — hm > —-=——= * 7 ΊΤ Λί^» ¿-», (ί - 2«Τ)2 + y2

APPROXIMATE HILBERT TRANSFORM OF PERIODIC DISTRIBUTIONS 237

Clearly for a fixed y Φ 0, q(t, y) £ Ρ2τ- It is shown that lim^o* Fix, y) = Hf, the Hilbert transform of the periodic distribution / , in the weak topology of PL:

= lim / i

= Hf.e ve e P2T

lim F{x,y).d(x) = lim / F(x,y)6(x)dx y—¥0+ -. . - .

Our result is used in finding the Harmonic function u{x, y),y > 0 which is periodic in x with period 2τ such that u{x, y) —► 0 as y —» °° uniformly V J C G S and

lim Μ(Λ, y) = /

where / is a periodic distribution with period 2τ.

7.8.1. Introduction to Approximate Hilbert Transform

If / £ W, p > 1, then its approximate Hilbert transform

j_ r f(t)(x-t)dt 7Γ J_a (x - 02 + y2 (HfKx)

in V sense as well as a.e. sense [99]. The main objective of this chapter is to find an analogue of this result for the space of periodic distribution PL and use our result in solving an associated boundary-value problem. To this end we define a function q : R X R -> R by

N 1 t - 2m

n=-/v

= l ' y - V ' ~ 2 η τ π (i2 + y2) /v—» 7Γ ^ (r - 2/IT)2 + v — . , y 2

n#0

Clearly q(t, y) is infinitely differentiable with respect to each of the variables t and y, and each such partial derivative is a continuous function of t and y over the interval [—τ, T], even on the real line. We define the approximate Hilbert transform Fv(x) of the periodic distribution / by

and prove that

Fv(x) = {f(t), ξ(ί)ς(χ - t, η)> (7.49)

lim Fn(x) = f{x) in PL 71—►<? +

That is,

ra+2r lim / Fvix)<pix)dx = Qif,to) V<p £ Ρ2τ (7.50)

Here Hf indicates the Hubert transform of the distribution / , which is periodic with period 2τ, and a is a fixed real number. Taking a = — τ, we get

lim / F^x^Wdx = (Hf, ξ(()φ) V 0 will also be proved.

Our result will be very useful in solving the boundary-value problem

uxx + Uyy = 0, v > 0, x G R

u(x, y) is periodic in x with period 2τ and

lim u(x, y) = f in the weak topology of Q2T (7.51) y—*o+

where / is a periodic distribution with period 2τ.

7.8.2. Notation and Preliminaries

The letter / throughout this chapter stands for a periodic distribution with period 2τ, T > 0 unless stated otherwise. The testing function space P2r is the same as defined in [ 12] and [ 110]. It consists of infinitely differentiable periodic functions with period 2T, T > 0, defined on the real line R. The topology over the space Ρ2τ is defined by the separating collection of seminorms {γ*}™=0 where

-y,(e) = sup|0<*>(/)l (7.52) t

Therefore a sequence {0„}"=1 tends to zero in the space Ρ2τ if and only if for each k = 0 ,1 ,2 ,3 , . . . , γ*(0ν) —> 0 as v —♦ °o. The space Ρ2τ is closed under convergence. As a matter of fact Ρ2τ is locally convex sequentially complete and a Hausdorff topological vector space [101]. Throughout this book D stands for the space of infinitely differentiable complex-valued functions with compact support. We now prove

Lemma 2. Let θ(ί) E P2T. Then VJT G [-τ, τ],

|0(JC)| < A \JT \θ(ξ)\άξ + JT \θ'(ξ)\αξ

where A is a constant independent of Θ.

Proof. Our result follows by using a slight modification of the technique used by A. Friedman in [3, p. 83], and it goes as follows:

APPROXIMATE HILBERT TRANSFORM OF PERIODIC DISTRIBUTIONS 239

Set S = | j , where T is a distribution of compact support. Now

/9 rfT T = d-kT = —[h(x)*T] = h(x)* —

dx dx

or

T = h(x)*S

Take a G D such that a = 1 if |*| < 5 and a = 0 if |JC| > 1. We assume that T is a regular distribution generated by a(x — y)(f(y) for a fixed real x lying in the interval [ — τ, τ].

The function θ(*) €Ε /"2T; also θ(χ) ε £. The regular distribution generated by a(* ~ y)0{y) also belongs to £', the space of distributions of compact support. So

i*«5[° a(x - y) 6(y) = / Λ ( £ ) ^ [ « ( Λ : -y + t)6(y- ξ)] άξ

Now take y = x. We get

a(0) Ö« = y Λ(£)^ [α(ξ) θ(χ - ξ)] άξ

Therefore

\Θ0 ) | < β / Τ[\θ(ξ)\ + \θ\ξ)\]άξ J-lcT

Here k is a sufficiently large positive integer such that the support of a is contained in [—kj, kr], Since Θ is periodic with period 2τ, we have

leool s ω [ £ |e(f)|rfí + / \β\ξ)\άξ D

Lemma 3. Let Θ G P2r. Then for p > 1, there exists a positive constant C indepen-dent of Θ such that

IÖl s C [HAH, + ||0<»||ρ

where

II011, ■[/· I ewl'A 1/P

Proof. This is an immediate consequence of Lemma 1 and the Holder's inequality. Note that if Θ e />2τ, then θ(*> e Ρ2τ for each k = 1,2,3 □

Lemma 4. Let us define the sequence of seminorms {β*}"=0 over the testing function space /*2T by

Α(β)= (J7_my\dt\ UP

for an arbitrary (fixed) p > 1, and let {γ*}™=0 be the sequence of seminorms defined by (7.52) over the space /gr-

illen the topology generated by the sequence of seminorms {γ*}*=0 over the space P2T is equivalent to the topology generated by the sequence of seminorms {/3*}"=0 over the space P2T.

IUP

'dx (2τ)1/ργ*(0)

Proof. Let 0 £ Ρ1τ. Then

ft(e(t)) = jT_Wk\x)\p

Using Lemma 2, we get

|0«>U)l =£ A i f \Θ«\ξ)\άξ + j T \θ«+]\ξ)\άξ

Therefore

sup |e w U) |<(2T) l / , ?

(7.53)

(f \βΡ\ξ)\'άξ\ " + ( T \θ^\ξ)\"άξ\ UP

where

or

1 + I = i p q

γ,(θ)<(2τ)1/"[/3λ(β) + β^ ι (β)] (7.54)

The result now follows in view of (7.53) and (7.54). The topology generated by the sequence of seminorms {/3*}"=0 over the space Ρ2τ

is the same as the topology generated by the sequence of seminorms {/3¿}™=0, where

A(fl) = max[ft,(e).ft(e) ft(0)]

[110, p. 8]. Therefore there exists a constant C > 0 and γ ^ Ο satisfying

Ι/.ΘΙ < Cjfyfl) V0eP2T

If it is not so, we can find a sequence {θν}™=χ in / V satisfying

| / . e j > vßi(0,)

A STRUCTURE FORMULA FOR PERIODIC DISTRIBUTIONS 241

or

/ νβ'ν(θ¥)) > 1 Vv = 1,2,3,...

Now put ψν = -fffl .. Therefore

|/.ψ„| > 1 V v = 1.2,3,...

For each k < v,

β*(Ψ,) s #(ψ„) = - -» 0 asv-^oo

For each k = 0,1,2,3

ft(«M s 0Í(«M -» 0 a s v - o o

Therefore by Lemma 4 we see that φν —► 0 as v —»°° in /,2τ. But

|/.ψ„| -» 0 as v -^ a>

a contradiction. This contradiction establishes our assertion. D

7.9. A STRUCTURE FORMULA FOR PERIODIC DISTRIBUTIONS

Theorem 7. Let / e Ρ2τ, and \f2r be the space of periodic functions with period 2τ which are V integrable over an interval of length 2τ. Then for each fixed p > 1, there exist measurable functions /* e Lj,., £ = 0,1,2 v such that

/ = ¿(-l)*D*/* *=o

That is,

/■e=/¿(-D*D*/*.íe\

= Σ / m<f\t)dt k=0J~T

ve e />2τ, ξ e i/2r

Proo/. Let q > 1 such that ± + ± = 1. If Θ e /»2τ, then Θ € Dq . That is, 0 is a periodic function with period 2τ such that f_r \d(x)\qdx < <».

Then, as stated before, there exist a constant C > 0 and a nonnegative integer γ such that

l/.ei s cß;(»)

or

I/.0I < C [0o(0) + β,(β) + · · · + βΎ(θ)}

Therefore there exist measurable functions /„ £ L£T such that

f.e = J2fa.Dae = ¿(-D)°/aÖ V0 £ Ρ2τ α=0 ο=0

The periodicity of each of f¡p follows by virtue of the fact that each element of the dual of P2l can be identified as a periodic distribution with period 2τ, namely each fjp £ LjT. This proves our claim. D

Corollary 2. If / £ PL·, then there exist measurable functions /„ such that flr \fa(x)\ dx < oo and such that

/ = ¿(-D)e fa «=0

The result follows in view of Theorem 3 and the fact that if fa £ L P [ - T , T], then faeV[-T,T],p>\.

Definition. If / is a continuous linear functional over the space P2T, then its Hubert transform as a continuous linear functional over Ρ2τ is defined by

Hf-e=-f-H9 V0 £ P2T

Thus, if / £ PL·, then we define its Hubert transform Hf by

<///, £(r)0(O> = if. ~ξ(ί)ΗΘ) V0 £ P2T

Clearly we see that Hf £ PL·. For if we assume that 0(f) is an element of Ρ2τ generated by a φ £ D through (7.22) then

(Hf, φ(ί)> = Hf ■ Θ = (Hf φ(ί + 2τ)>

Theorem 8. Let F(x, y) be the approximate Hubert transform of a periodic distribu-tion / £ PL·, T > 0, defined by

F(x,y) = Fy(x) = / / (*) . £(f) i .(jt - 0 2 + y2

= (f(t),mq(x-t,y)) where

1 A (jc-2/ιτ) q(x,y) = hm — > —-=——=■ HK " N^OO TI Δ-. (χ- 2nr)2 + y2

Then ^,L„F(x, >) are all continuous functions of (x, y) for each m, n = 0,1,2,3 and

-F(x, v) = ( / (f) ξ(0——α(χ -t,y)) (7.55) dxmdy" \ dxmdyn

lim F{x,y).e(x) = lim / F(x,y)e(x)dx (7.56) >-o+ >^o+ J_T

= {Η/,ξθ) V 9 6 P 2 T

where Hf is the Hubert transform of the periodic distribution / .

Proof In view of the proved structure formula for / e PL, we see that

F(x.y) = ( ¿ ( - l ) e D e / a ( 0 . i ( 0 - Hm £ r - ^ ~ 2 " T ^ 2/ιτ)2 + v2

or

FU, y) = Σ / /« W * * - '- ) * (7.57) a=0

Since .£ Λ <?(*, y) is a continuous function of (JC, y), ¿Ur"«?y

3ΛΙ + Π 7 ,-τ

-F(jc,y) = 5 1 / fAODfD^D^x - t,y)dt (7.58) dxmdy

Therefore (7.55) follows in view of results (7.57) and (7.58). To prove (7.56), let θ(χ) G Ρ2τ· Then

F(x,y)Mx)= Í F(x,y)e(x)dx

F(*,y).0(x) = ] T / / fa(t)D?D?Dyq(x-t,y)dte(x)dx (7.59) a=0·

Since q(x — t, y) and all its partial derivatives are smooth functions, we can switch the order of integration in (7.59). Therefore, for a fixed y > 0,

F(x,y)Mx) = / / (*) . f (0 f Φ - t,y)e{x)dx\

Our result will be proved by showing that

lim / q(x -t,y)6(x)dx = -ΗΘ >->o+7_T

in the weak topology of PL.

Now

ΐή i q(x-t,yMx)dx= j q(x - t,y)e(k\x)dx

and

Df(//0)(/) = (Heik))(t)

Our result will be proved by showing that

lim / q(x,y)d(x - t)dx = -ΗΘ >-0+ J-r

uniformly for all / in the interval [ — τ, τ].

1 χττ 2τ 2 T

= Hm y x - 2ητ 1

n = -N L

N

(x - 2 / IT) 2 + y2 (x - 2ητ)

_ , , 2 = lim > -

N~" „ΓΤ'Λ, (X - 2ητ)[(χ - 2ητ)2 + y2]

[y2x + xi + llxn2·!2]

n = -N

„2

x(x 2 + v2) ^ 7 x1 - 4χ2τ2)[(χ - 2nrf - y2][(x + 2ητ)2 + y2]

(7.60)

The second term in the right-hand side of (7.60) —► 0 as y —> 0 uniformly for — T ^ x ^ T.

Therefore, in view of Lemmas 3 and 4 and Theorem 8, our result will be proved if we simply show that

lim Γ / L· 2J(x-t)dx = 0 V0We/>2T >-o+ J_T x(x2 + y2)

uniformly for all t €Ξ [—τ, τ]. Now

„2 (P) r f_

J-r XU2 + y2) e(x)dx = (/>)

Jo θ(χ) - θ(-χ) y2

x x2 + y2 dx

Define a function φ(χ, t) as follows:

g(jc - Q - g ( - s - t) <f>(x,t) = . x Φ 0

= 2Θ'(-/) , χ = 0

Clearly ψ{χ,ί) is infinitely differentiable over the interval [ —τ, τ]. Therefore

P I < L· 2J(x~t)dx= f <p{x,t)^—^dx (7.61) J-T x(x2 + y2) J0 x2 + y2

Denoting each of the expressions in (7.61) by /, we get „2

l/l < 2 sup |e'(jt)l / - T S j t « T JO x2 + y2 dx

IT < 2 sup |e'<jr)|y— — 0

- T R A T S T -¿

as y —► 0+ uniformly V/ G [ — τ, τ]. This completes the proof of our claim. □

7.9.1. Applications

We will now apply our result in solving a boundary-value problem: Find a harmonic function u(x,y) in the region y > 0 such that u(x,y) is periodic in x with period 2τ and that

lim u(x,y) = / in£>2T y—»0 +

and

"(-*. y) = o(l) as y —► oo uniformly V* G /?

Here / is a periodic distribution with period 2τ and (?2τ = {Θ : 0 G /^τ such that / Ι τ θ(') A = 0}. The topology of Q2r is that induced by Ρ2τ. Take

«(*,?) = (-(«/)(0. i (0 1 x - f

= -("/)( ')

TT(JC-Í )2 +y 2

JC - f

TT(X - t)2 + y2

(7.62)

(7.63)

where

[0(0] = Σ W - 2nr)

By Theorem 3, it can be shown that

lim u(x,y) ■ θ(χ) = (-Η2/,ξθ)

= {f, ~ξΗ2θ)

-('■(·- s/>*)«>

It can also be shown that

<«(jt.y).í(jc)e(jc)>= / u(x,yMx)dx (7.64)

θ(χ)άχ) (7.65)

Note that the range of integration in the right-hand-side integral in (7.64) can as well be taken from ητ to (n + 2)τ.

Letting y -» 0+ in (7.62), we get

lim (u(x, y), ξ(χ)θ(χ)) = (f, ξ(χ)θ) V0 E β2 τ

The fact that u(x, y) = o(l) as y —* °° can be proved by using Holder's inequality. We now solve the above problem slightly differently.

Solution. Given

Uxx + uyy = 0, y > 0

"(*. y) = o(l), as y —» oo uniformly VJC E /?

lim κ(*, y) = / in Q2T where / E PJT y—>0+

That is, / is a periodic distribution with period 2τ. Take

u(x,y) = - ( / ( f ) .£(0 Hm V ^ . , , ' π V ΛΤ-» f - „ (f - JC - 2/IT)2 + y 2 /

As in the previous exercise, we take

u(x,y) = ( -(Ηη(ί),ξ(Ο- lim V - — - — ' "T, , ) 7 \ J ir ΛΤ-» ^ (χ - ί - 2ητ)2 + y2 /

= (f(t).i(O- Hm Υ^ 7 ~ ,-, , - ) \ J i V 1 , Ν - " ' „ Γ ! ' Λ , (* - Γ - 2/ιτ)2 + y2 /

As before, Μ is a harmonic function of x, y in the region y > 0 and M(JC, y) —> / in /»2T as y —► 0+. The fact that u(x, y) = o( l) as y —► °° uniformly V jr E /? can also be proved by using the structure formula for / and Holder's inequality. The harmonicity of U(x, y) can be proved by direct differentiation. The fact that lim>_ot V(x, y) = f in Q'1T can be proved by using the structure formula for / .

EXERCISES 247

EXERCISES

1. Let / £ PL and H be the operator of Hilbert transformation over PL. Prove that

\ 7T ΛΓ—°° ¿—' (χ — t — 2ητγ + yl /

= (m lira i(0— E 7 , ,Λ ,2 , 2 V Í £ P2r \ AÍ-.CO 7Γ ¿—' (x - t - Ίττ/ητγ + y1 I

2. Prove that

lim / / ( ί ) , - ξ (Ο Hm V —-, A = f y^o+yK π *-·» ¿-*N (n - t - 2nir)2 + y2 / J

in the weak topology of PL. 3. Find a harmonic function u(x, y) that is periodic with period l in the region y > 0

such that lim^o* u(x,y) = ΣΓ=-°° δ(ί - n) in P[ and «(*,)>) = o(l), y —> °° uniformly Vx £ Ä. Verify your result.

4. Prove that every element of the dual space of Q2T can be identified as a Schwartz distribution with period 2τ.

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INDEX

Absolutely convergent, 221 Absolutely integrable, 44, 122, 123, 137,

233 Absorbing set, 1,4, 15, 16, 18, 20, 51 Analytic function, 54, 57, 58, 62, 67, 68,

69, 77, 78, 85, 104, 106, 108, 110, 111,118, 121,165,167,168, 170,217

at a point, 55 in a disc, 55

Approximate Hubert transform, 101, 102, 197, 198, 200, 236, 237

of periodic distribution, 236, 242 Associated Riemann Hubert problem, 170 Automorphism, 184

Balanced, 14, 16, 18,20,51,52 Balanced set, 15 Ball in a NLS, 19, 33 Balloons, 22 Banach space, 14, 52 Basis for topology, 22 Borel measurable, 8 Borel measurable function, 8, 187 Borel measure, 187 Boundary value problem, 169, 237, 238,

245

Calculus of distribution, 29 Canonical isomorphism, 43 Cauchy integral, 54, 64, 119 Cauchy Theorem, 56, 61, 79, 165 Characterization of £'(R"), 33 Characterization of distribution of compact

support, 32

Characterization of LCS, 19 Classical Fourier transform, 89, 178, 204 Classical Hubert transform, 91, 92, 96,

110,113, 116, 118, 124, 153,160 Classical inversion formula, 127 Classical «-dimensional Hubert transform,

156 Compact support, 3, 22, 187, 192, 222 Complex Hubert transform, 202, 204, 212 Compatible topology, 13 Conjugate Harmonic function, 92, 107,

108, 109 Contraction, 13 Convergence of distribution, 29 Convex set, 14, 19, 42, 51, 52 Convex hull, 52 Convolution, 91

of classical function, 11 of distribution, 34, 36 of function, 7

Convolution operator, 113 Countable family of separating seminorms,

195 Countable local base, 18 Countable set, 122 Countable union, 132 Countably multinormed space, 139

Dense, 126, 131, 139, 141, 143, 152, 159, 160,163, 168, 179, 180, 188, 207, 208

Density, 191,201 Differential operator, 38 Differentiation of the Fourier Transform,

17

255



256 INDEX

Dirac δ-function, 27 Dirichlet boundary value problem, 107,

109,149,151,194,213 Direct product, 34

of distribution, 34 Direct sum, 190 Distribution, 26, 34,48, 91, 94, 96, 114

of compact support, 32,46, 168 of slow growth, 41

Distributional boundary condition, 157, 204

Distributional convolution, 97 Distributional differentiation, 29, 31, 157,

204 Distributional Fourier transform, 215 Distributional Hilbert problem, 111, 138,

215 Distributional Hilbert transform, 83,96,

131 Distributional representation of

Holomorphic function, 265 Dispersion relation, 86 Divergent part, 28 Double Hilbert transform, 184 Duality, 77, 229 Duality results, 11 Duality Theorem, 204 Dual Space, 231

Elliptic crack, 84 Embedding, 24 Entire function, 49, 59, 88,112,192 Eigenfunctions, 113, 214, 215 Eigenvalues, 13, 214, 215 Equivalent neighbourhood basis, 14 Equivalent topological base, 14 Exponential growth, 192

Factorization, 111 Finite order derivative, 32 Finite Hilbert transform, 80, 82, 84 First countability axiom, 15 Fourier integral operator, 7 Fourier reciprocity relation, 7 Fourier Series, 218 Fourier transform, 40,43, 44, 45, 52, 89,

93, 114, 121, 122, 123, 125, 126, 131, 151,178,181, 182, 188, 192, 193, 195,206,207,208,211,215

of distribution, 1,44,48,49, 50, 51 of functions, 1, 3,46, 52

Fourier transformable, 5, 212 Fourier transformation operator, 9, 132 Fourier transformation of Schwartz

distribution, 1,3,36,52 Frechet space, 42,52, 131, 231 Fubini theorem, 9,150, 163, 179, 181, 187,

199,200 Functional, 109,115, 180,225, 229,230,

232, 242 Function space, 49, 228, 229

Generalized function, 26, 52, 113, 121, 125,128, 146,161, 182,215

Generalized Hilbert transform, 97, 99, 116, 145, 148,160,214

Generalized (n + 1 (-dimensional Dirichlet BVP, 149

Generalized Parseval identity, 76 Griffith Crack, 85

Hadamard, 28, 29 Hahn-Banach Theorem, 43 Hausdorff, 17, 18,52 Hausdorff topology, 18 HausdorffTVS,42,94, 126 Holder's inequality, 10,95, 195,239,246 Holomorphic function, 85, 110, 115, 118,

163,167,172,174,175,205,206,209 Homeomorphism, 15,44, 45, 50, 51, 89,

94,97, 114, 115, 118, 126, 131, 143, 146, 154, 160

Harmonic function, 90, 107, 108, 149, 151, 192, 237, 245, 246, 247

Hilbert inversion formula, 71 Hilbert problem, 82,85,111,112,170,

173, 174,177, 178,215,216 Hilbert reciprocity relation, 218 Hilbert transform, 66, 67, 69, 70, 75, 76,

81, 84, 86, 89, 90, 91, 92,93, 96, 107, 113,114,115,116, 121,122, 123, 125, 126, 127, 128, 131, 133, 135, 145, 151, 152, 153, 154, 159, 169, 170, 178, 180, 181, 182, 183, 184, 186, 198, 200, 206, 207, 208, 215, 217, 218, 219, 220, 221, 228, 229, 230, 231, 232, 233, 234, 237, 238, 242, 243

INDEX

of distribution, 156 of Schwartz distribution, 156 of test function, 142 type operator, 209

Hubert transformable, 212, 229 Hubert type transform, 178, 183

Inductive limit, 24, 130 Inductive limit topology, 31 Integrable singularity, 71, 73 Integral equation, 72, 87, 99, 117, 148, 236 Integral transform, 3 Inverse Fourier transform, 1, 45,49, 50, 89 Inversion formula, 138, 152, 153, 155, 175,

176, 184, 217, 219, 230, 234, 235 for finite Hubert transform, 80 for Fourier transform, 3 for Hubert transform of periodic

function, 68, 76 Inversion theorem, 97, 124 Isometric isomorphism, 180 Isomorphism, 97, 145

Lebesgue density theorem, 191 Lebesgue measure, 9, 185, 191 Lebesgue measurable set, 183, 185, 190,

191, 198 Lipschitz condition, 56 Locally convex TVS, 18, 20, 21 Locally convex Hausdorff TVS, 20, 22, 94,

126, 131, 133, 154, 195 Local base, 21,22 Locally convex spaces, 21, 24 Locally integrable, 26, 121, 125, 233 Locally integrable periodic function, 226,

233

Mean Value Theorem, 146 Measurable (Borel), 8 Measurable function, 195, 228, 241, 24 Measurable (Lebesgue), 8 Metric, 18 Metrizable, 18,21,22, 141 Minkowski functional, 16, 18,21 Morera's theorem, 59 Multinormed, 122

Neighborhood of a point, 13, 56, 64, 68, 87 Neighborhood basis, 32, 52

257

Norm, 149, 188 Normable, 19, 21 Norm topology, 152 n-dimensional approximate Hubert

transform, 199 n-dimensional complex Hubert transform,

202 n-dimensional Fourier transform, 131 n-dimensional Generalized Hubert

transform, 160 n-dimensional Hubert problem, 174 n-dimensional Hubert transform, 131, 138,

142, 152, 156, 184, 185, 192 n-dimensional truncated Hubert transform,

185 n Riesz transforms, 189, 190

One-dimensional Hubert operator, 201 Open set, 12 Open neighbourhood basis, 22 Orthants, 192, 193

Paley Weiner Theorem, 192, 193, 212 Parseval type relation, 89,93, 114, 151 Period, 68, 217, 218, 219, 220, 221, 239,

224,225, 226,228,229, 232, 233, 236,237,238,239,241,245

Periodic distribution, 217, 223, 224, 225, 228, 230,232, 236, 237,238, 242, 243, 245, 246

Periodic function, 66,67,68,69,71, 87, 217, 218, 219, 220, 221, 222, 225, 228, 229, 233, 234, 237, 238, 239, 241,245

Periodic testing function, 228 Plemelj formula, 56, 60, 62, 63, 64 Poincare Bertrand formula, 66 Poisson kernel, 149 Principal value, 54

Reciprocity relation, 76, 77, 78, 109, 218 Regular, 76, 77 Regular Borel measure, 187 Regular distribution, 26, 29, 36,41,96,

102, 116, 121, 204, 223, 226, 239 Regular generalized function, 125 Regularizaron of distribution, 39 Regular tempered distribution, 188, 207 Removable singularity, 76

258 INDEX

Riemann Mapping Theorem, 57 Riemann Hubert Equation, 171 Riemann Hubert problem, 84, 170, 171 Riesz Representation Theorem, 187 Riesz transforms, 189, 190

Scattering amplitude, 86 Schwarte distribution, 90, 107, 114, 116,

146, 160,247 Schwarte generalized function, 178 Schwartz space, 152, 184, 192, 194 Schwartz Theory of distribution, 12 Schwartz Tempered distribution, 126 Schwartz testing function, 52, 113, 139,

152, 168, 178, 179, 187, 198, 222 Schwartz testing function space, 19, 136 Schwartz testing function space of rapid

decent, 137 Second countability axiom, 15 Seminorms, 17,40,42,93, 138, 139, 180,

236, 239, 240 Separating collection of semi-noimess, 22,

154,229,238 Sequentially complete TVS, 22, 94, 126,

132, 154, 165, 195 Singular differential equation, 125 Singular distribution, 27, 111 Singular equation, 117 Singular integral equation, 58, 138, 151,

217,235 Singular integro-differential equation, 115 Singularity, 76, 87 Space of distribution, 239 Space of periodic distribution, 236, 237 Space of ultra distribution, 57 Strict inductive limit topology, 132 Strong topology, 97,146 Structure formula, 139, 150, 196, 203,

246 Support of function, 27 Support of a distribution, 37

Tensor product, 34 Tempered distribution, 41, 42, 44, 45, 89,

195 Test function, 139, 143, 222

Testing function space, 40, 120, 131, 132, 136,138, 153, 158,180, 193,208, 222,231,238,240

Testing function space of rapid descent, 195

Theory of distributions, 12 Titchmarch's inversion formula, 88 Titchmarch theorem, 7 Topological bases, 14 Topological space, 12, 22 Topological vector space, 13, 14, 15,51, 52 Topology, 12, 40,41, 50, 52, 93, 114, 115,

118,121, 122, 124, 126,130, 131, 132,133, 137,138, 139,151, 152, 154, 159, 161, 178, 179, 180, 195, 208, 229, 236, 238, 240, 245

Translation invariance, 9 Translation mapping, 15 Truncated Hubert transform, 183, 185 Truncated operator, 190 Two dimensional Hubert transform, 176

Ultradistribution 51, 52, 117, 122, 125, 126, 127, 128, 132, 133, 135, 160

Uniform continuity, 3, 136 Uniformly bounded, 1 Uniformly convergent sequence, 140, 141,

164, 196,221 Unitary, 222, 223

Vector space, 52 Vorticity vector, 83 Vortex, 83 Vorticity distribution, 83

Weak distributional sense, 179 Weak limit, 82 Weakly complete, 30, 227 Weak topology, 45, 89, 97, 101, 102, 107,

108, 111, 146, 178, 181,198,237, 238, 243, 247

Weirstrass Approximation Theorem, 139 Wing theory, 82

Z-transform, 218 Zero ultradistribution, 128, 135

NOTATION INDEX

B, 15 Bx, 14, 15,22 By, 15, 16 B„, 18 B, 19, 182, 183 Βψ,22 β, 186, 187, 188, 192

C;£, 185 C, 13,14, 16,20, 163, 187, 167 C \ 177 C(m), 21 C*"0, 21 C", 21, 27, 39, 40, 93, 95, 114, 115, 121,

126, 130, 136, 137, 150, 152, 153, 159, 167, 179, 187, 194, 195

C~, 21 Co, 21, 23, 25 CJ", 22, 23 (Cr

m(n))', 22 C, 30,57 C 52 C„,52 Cf, 55, 61, 62, 63, 64, 88 C , 55, 63, 88 Cp,95 C , 138, 163, 164, 167, 192, 202, 205, 212 C;, 184 Cn

p, 184, 201 c;:£, íes C'p", 185

D, 3, 11, 23, 25, 26, 28-34, 36, 39,41,49, 50-52, 86,90-91,94, 113-116, 118-121, 117,139-142,152, 158-164, 166-168,179,180, 187, 188,200,207,222-227,339

D \ 29, 30, 31, 33, 34, 39,41,49,51,86, 90-94, 112, 114, 116-118, 151, 152, 160,161,184, 215,223,227,228

dE, 191-192 Dk, 23,24,25, 30, 31, 34,35, 36, 39 V,,T, 35, 36 D,'T, 34, 35 O'k, 34 I T , 39 ©[,,89-100, 101-104, 106-114, 116,

138-139, 145-148,150-153, 156-159,163,171-175,176, 184, 192-195, 197, 203-206, 208-210, 212-215

T)^, 96-98, 100-104, 108, 118-119, 121, 139, 146, 147, 150

1)^,93,94,96,98, 100, 111, 115-116, 108,138-139,141-146,153-155, 159-165, 167-168, 178-181, 194-195, 208, 216, 212

DP, 25 D^ ' ,94 (27^)',94, 113, 116 (2V) ' , 94 Ox,,, 156-157, 195, 204, 208 (DLP(W))', 168, 180-182, 198,212

259



260

Ον(_τ,τ), 229-230 Ό,ρ , 241

ST

£',32,33,39,41,46,53 £,8,13,17,32,33 £ ,8 £°, 191

£, 1, 2,4, 8,9, 163, 170-172, 174-177, 181,192-197, 202-206,209-215, 221-222,236-237, 242,243

J, 1, 2,4,5,9,12,43-46,48, 51, 89,93, 121-134,178-180,182, 198, 208-213,215,207-208

/"' , 13,31,32 / * g, 7,8,9, 10,11,36-39,48,52 J-«, 1,44,45,89,137 F+, 54-56,61-04, 69, 71-73, 75, 87, 170,

172-175, 215 F-, 54-56,62, 65, 72-73, 87,170,

173-175,215 F++, 171, 173, 175-177, 205-206,210,

211,216 F—, 171, 173,175-177, 205-206, 210,

211,216 £+_, 171,175-177,205-206,210,211,

216 F- + , 171,175-177,205-206,210, 216 Fp, 54, 55,60-64,88 £η, 100-102, 237-238 JH, 80-82,93 Fai, 193,194 /+ ,104-106 / , 1 0 6 - 1 0 7 / M 0 6 £±±±, 177 F+++, 172, 173, 177 F 172,173,177 £+-_, 172 £-+_, 172 F—+, 172 £ + + - , 172 £ + - + , 172 £-++,172 £±,171 £„,, 173, 205, 210, 212-214

G+,65, 113 G-, 65, 113

NOTATION INDEX

Hf, 66, 68, 70, 75-79, 88, 90-93, 107-109,114,127,152,155, 157-159,170-172,175,183, 202-203, 209-210, 217, 220-221, 229-231, 233-234, 237-238, 242

H2,66,68,76,88,92,96-97,99,114,116, 118,121,124, 125,127,138,142, 143, 145, 147-148, 152-153, 160, 178, 184,214-215, 230-232, 234, 245

«' ,86, 114-118, 159-160,212 //"', 94, 96-97, 126, 143-144, 156, 232 //„, 100,102-103 //*, 124, 125, 145 Jfp, 107-110 Jf, 110 H, 113-117, 119-121 H'2,145 Ηε, 185-192 Hoi, 205 Z/,,218 / / Ä , 2 1 8

im, 106-107, 113, 119, 165-167, 174-175, 177, 202

/ ,67 /2 ,68 /*,95 K+, 110, 111,113 A:-, no, in

14.,219 ¿ ' ,2 ,3 ,4 ,6 ,8 ,9 ,10 ΖΛ4.6.93, 104, 178-179, 192,

206-207 ¿",43,187,188 U, 3,7,10,11,66-67,76-77, 81,92-93,

96,99,101, 103-106,107,113,116, 138-139, 145, 148, 150, 152-153, 171-173,175, 177-179, 181-195, 197-205, 207-209, 213, 229, 237, 241-242

¿«,7,10,11,149,197 L\r, 219, 221,228-230,242 ¿,60 V, 42-43, 292 L\r, 229 ¿+, 59, 71-75 ¿_, 59, 71-75

NOTATION INDEX 261

M+,58,59 Λί-,58,59 με, 16, 17

N, 91, 93-94, 101, 103, 118, 121, 127, 129, 130, 133-134, 154, 195

/V+,60 N-,60

o(l), 175, 177, 246-247

P, 17 P[, 247 Pp. 17 Pf, 28, 29 P¡T, 223, 226-232, 236-237, 241-243,

246-247

Q'2T 231, 245-246 $ „ . 2 3 5 β2, 236

R, 2, 3, 5, 6, 7, 9, 10, 11, 14, 32, 35, 38, 46^17,91-93,95, 104, 115, 121, 126, 140, 142, 149, 163, 171-172, 174, 177-179, 183, 190, 192, 198-200, 205-207, 220

R2, 8, 9, 10, 34, 35, 52, 95, 172, 175-176, 193, 199,205,211,216

R\3,4, 11,12, 14,21,23-26,29,32-33, 40-45, 49, 51-52, 89, 91, 131-135, 138-150, 151-168, 171, 173-174, 177-210,212-214

Re, 105, 111 R, 18,225,237,246-247 Λ+,14 R("+,) + , 149, 150 R<"+1), 151 R2", 195 R3, 169 Rl"-*\203

S, 40-45, 47, 52, 89, 126-127, 130-132, 137, 168, 178-182, 193, 195, 198, 207-208, 215, 239

5', 41-12,44-^5, 47, 51, 89, 127, 131, 132, 168, 180, 181, 188, 195, 207

S-,65 S+,65

S0 -si SP - 6 5

sgn, 113, 124, 126, 127-129, 133, 136, 178-179, 181-182, 194, 198, 207-209, 211

Si, 126, 128-129,130-131 S;, 130, 132 So, 130, 133-134, 137, 178-182, 198,

208-210,212-213 S^.S^.S^.-.S^ 130 S„ 130, 132 S[, 131 5η,,133 S¿, 178, 181-182, 194, 208-209,

212 S + - , 293, 205, 211 S+ + , 193,205,211 S_ + ,193,205 S-_, 193,205,211 Sait, 193, 194

W+, 56-59 W-, 57-59

Z, 49-51 Z,, 126, 127, 130, 131 Z{, 127-131 Zo, 132, 133, 135 Ζη, 130, 132 Z¿, 132-135 Ζ'Ψ 130 Z„, , 132-133 Z' ,51, 135

11112,6 INI,, 7,10, 11,196 II Up, 7, 10, 11,95,113, 116, 145, 152, 153,

155, 160, 164, 168, 180, 183-184, 186-187, 190-192, 196, 198-203, 229, 239

II Hi, 8, 9, 10, 11 II ||, 13,14,27, 30, 31,47, 138,152,188,

229 II I U R . , . 42 II II*, 122, 124 II lip-, 121 II I U 150 II II«, 149 II l h „ 199 II l k „ 199

262 NOTATION INDEX

< >, 11, 22, 26-38,41^2, 44-45,47-48, 77, 89-92,96-98, 100-110, 112, 114, 116-117, 124-125, 127-129, 133-134,139,146,150,156-157, 160-161, 171-172, 174-177, 180, 192-193, 195, 204, 207, 223-226, 230-232, 236, 2420243, 245-247

Ψκυ., 26 <Pvj,„ 26

»«LM·40

7ΪΜ· 4° TWIOI 41 γ+,110-112 γ_,110-112 %α. 180 ΧΕ, 190 * * . 191 ©,190

1 + , 28,29,36

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