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Integral transformations and spaces of type S
Citation for published version (APA):van Berkel, C. A. M. (1992). Integral transformations and spaces of type S. Technische Universiteit Eindhoven.https://doi.org/10.6100/IR376185
DOI:10.6100/IR376185
Document status and date:Published: 01/01/1992
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INTEGRAL TRANSFORMATIONS AND
SPACES OF TYPES
INTEGRAL TRANSFORMATIONS
AND
SPACES OF TYPES
PROEFSCHRIFr
ter verkrijging van de graad van doctor aan de
Technische Universiteit Eindhove~. op gezag van
de Rector Magnificus, prof. dr. J .H. van Lint,
voor een commissie aangewezen door het College
van Dekanen in het openbaar te verdedigen op
vrijdag 10 juli 1992 te 16.00 uur
door
CORNELIS ADRIANUS MARIA VAN BERKEL
geboren te Sint-Oedenrode (NBr)
druk: wibro dlssertaliedrukkerlj, helmond.
Dit proefschrift is goedgekeurd door de promotoren prof. dr. ir. J. de Gra.af
en
prof. dr. J. Boersma
Copromotor:
dr. ir. S.J.L. van Eijndhoven
Aan mijn ouders
CONTENTS Introduction
1. Functions of rapid decrease
1.1 The Schwartz space S(JR.) 1.2 The subspace Seven( JR.) and the Hankel transformation Di11
1.3 The space S(JR.+) and the Hankel-Clifford transformation 1-t, 1.4 The Schwartz space S(JR'l) 1.5 Spherical harmonics 1.6 The space S(JR'l) and expansions in spherical harmonics 1.7 The Radon transformation on S(JR.q)
2. Gel'fand-Shilov spaces
2.1 The Gel'fand-Shilov spaces S~(JR.) 2.2 The subspaces S~,even(JR} and the Hankel transformation D:l" 2.3 The spaces G~(JR+) and the Hankel-Clifford transformation 1-t., 2.4 The Gel'fand-Shilov spaces S~(JR.q) 2.5 The spaces S~(JR.q) and expansions in spherical harmonics 2.6 The Radon transformation on sg(JR.q)
3. Fractional calculus and Gegenbauer transformations
3.1 The Weyl fractional operators 3.2 The Riemann-Liouville fractional operators 3.3 Rodrigues-type formulas for Gegenbauer functions 3.4 The Weyl-Gegenbauer transformations 3.5 The Riemann-Liouville-Gegenbauer transformations 3.6 The inverse Gegenbauer transformations 3. 7 Integral equations involving Gegenbauer functions
Bibliography
Glossary of symbols
Index
Samenvatting
Curriculum vitae
1
9
9 14 23 29 34 41 50
61
61 74 78 84 93 99
105
105 111 115 119 122 124 128
135
138
141
142
143
INTRODUCTION
Generalized functions are of great interest in several branches of mathematics and physics. Laurent Schwartz [45] has contributed importantly to the development of the theory of generalized functions (distributions). Schwartz 's construction of a Fourier invariant theory of generalized functions is based on the test-function space S consisting of all infinitely differentiable functions f(x) = f(x 17 ••• , xq) with the property
oh+···+1• f(x) sup lx~' ... x:• 1 1 I< oo, ki,li = 0,1,2, ... , j = I, ... ,q. xeR• oxl ... 0Xq0
Inspired by Schwartz's work, Gel'fand and Shilov [22] introduced and investigated some other types of test-function spaces, which they called "Spaces of type S". Their definition is the following. For non-negative numbers a 17 ••• , a 9 , the space Sa = Sa,,a2 , ••• ,cr. consists of all infinitely differentiable functions f on IR.9 with the property
I k, ko olt+ ... +l•f(x)l sup x1 ... x
9 1 1 xeR• oxl ... 0Xq0
< B Ak' Ak• kk' a, kk•a• - 1., ... ,1. 1 • . • q 1 • . . q ' kj,lj=0,1,2, ... ' j=l, ... ,q,
where the constants B1,, ... ,19
, A17 ••• , Aq depend on the function f. For non-negative numbers {317 ••• ,{39
, the space Sl3 = Sf3,,(32 , ... ,(3. consists of all infinitely differentiable functions f on IR.9 with the property
I k, k• (JI• + ... +lo f(x) I sup x1 ••• xq 1 1 xeR• oxl ... Oxq•
ki,li=O,I,2, ... , j=l, ... ,q,
where the constants Ak,, ... ,k., B17 ••• , Bq depend on the function f. The space S~ = S~::~·::.:~~ consists of all infinitely differentiable functions f on IR.9 with the property
I k, kq [JI• + ... +lq !(X) I
sup x1 ... x9 1 1 xeR• oxl ... Oxq•
kj,/j=0,1,2, ... '
j=l, ... ,q,
where the constants C, A1 , ••• , Aq, B1 , ••• , B9 depend on the function f. Obviously, the spaces Sa, Sl3 and S~ are subspaces of S. In the definition of the space Sa there are mainly constraints on the decrease of the functions at infinity, while in the definition of Sl3 there are mainly constraints on the growth of the partial derivatives ( olt+ ... +l. 1 ox~' ... ox~•) J(x) as 11 + ... + 19 ---+ oo.
If Oj ~ ai and jji ~ /3j for j = I, ... , q, then we have the inclusions
Sa c Sa , siJ c S 13 , sg c S~ .
2
For arbitrary a;,/3; 2::: 0, the spaces Sa and SPare nontrivial, i.e. contain not only the null function. The space S~ is trivial if and only if for j = 1, ... , q,
a;+ fJ; < 1 or (a;,fJ;) = (1,0) or (a;,fJ;) = (0, 1).
Fro~ a topological point of view, Gel'fand and Shilov [22] described the spaces Sa, SP and S! as a union of countably normed spaces. Thus they were able to define sequential convergence in the spaces Sa, SP and S!, such that these spaces became sequentially complete. Gel'fand and Shilov investigated their spaces of type S for the sake of solving the Cauchy problem for partial differential equations. For the functional analyst these spaces are interesting in themselves because of their rich structure. In each of the definitions of S, S0, Sl3 and S!, the supremum norm can be replaced by the norm in L2(/R"). Therefore, the theory of unbounded linear operators in Hilbert space can be invoked to give alternative descriptions. In the Hilbert-space approach the spaces of type S can be expressed as the intersection of so-called Gevrey spaces brought about by the self-adjoint operators Qi of multiplication by Xj, and the self-adjoint operators P; = iojfJx1 of partial differentiation, j = 1, ... ,q, in the Hilbert space L2(1Rq); see Van Eijndhoven [12] and Ter Elst [17]. Now the theory of locally convex topological vector spaces can be applied. Indeed, the Gevrey spaces are equipped with a well-described topology and so it is natural to endow the spaces of type S with the corresponding intersection topology. · It can be shown that sequential convergence in the sense of this intersection topology agrees with Gel'fand and Shilov's definition of sequential convergence. Let us mention some nice properties of the spaces of typeS. (i) The space So,o ..... o equals the space of all infinitely differentiable functions on JR'I with compact support. (ii) For a; > 0 , j = 1, ... , q, the functions fin Sa decrease exponentially: There exists t > 0, depending on J, such that
sup I exp(t lxtlt/a, + tix2l 1/a2 + ... + tlxqll/a9 ) f(x)l < oo. xERq
(iii) For 0 :5 (Ji :5 1 , j 1, ... , q, the functions in Sl3 have an analytic continuation into the complex space V. (iv) The class of spaces of typeS remains invariant under the Fourier transformation IF on L2 (JR'I),
IF(Sa) =sa , IF(SP) = Sp , IF(S~) = S$ .
(v) Evidently, the spaceS~ is contained in San SP. Kashpirovskii [33] showed that
s~ = s, nsP. (vi) The Hermite functions
tfon(x) (11'1/22"n!t1
/2 (-It exp(~x2 ) (d/dxt exp(-x2
), nE /No, x E lR,
constitute an orthonormal basis in L2(/R). So the products Wn(x) = lll=t tfon1(x;), nE INJ, x E JR:l, constitute an orthonormal basis in L2(/Rq). Simon [46] has proved that the space S can be characterized in terms of the expansion coefficients of its elements, with respect to the basis {w.. : n E /N6}:
3
The analogue for the spaces~ with Oj :::: !. j = 1, ... 'q, reads
s~ = u e L2(IR") : 3t>o
(exp(tn:/(2at) + tn~/(2a2 ) + ... + tn!/(2a•)) (!, 11'n)L2(R•)) E loo} ,
which was proved by Zhang [51] in the case q = 1. Such a characterization cannot exist for the other spaces of type S, because they are not Fourier invariant (see property (iv)). The space S~ is the analyticity space corresponding to the positive self-adjoint operator EJ=1 ( -d! fdx~ + xD1
/(2a') in L2(1Rq), cf. De Graaf [24], De Bruijn [3].
In this thesis we go into a further study of the spaces S, Sa, Sf1 and S~. The space S is called the Schwartz space and the spaces Sa, Sf1, S~ are called Gel 'fand-Shilov spaces. We only consider the case o1 = o2 = ... = Oq = a, {31 = {32 = ... = f3q = (3. We deal with several integral transformations (Fourier, Hankel, Radon, Weyl, Riemann-Liouville, Erdelyi-Kober). In particular, we are interested in finding new characterizations of the Gel'fand-Shilov spaces by means of these integral transformations.
Example 1: A function f on /R9 belongs to S" if and only if there exist A, B > 0 such that
and
ll(x~ + ... + x~)112 lF JIIL2(R•l < oo , 1 E /No .
From the properties (iv) and (v) of the Gel'fand-Shilov spaces, similar characterizations for the spaces Sf1 and S~ follow, see Chapter 2, Theorem 2.42. Starting point of our investigation was the well-known result that a function f E L2(/R9 )
has an expansion with respect to spherical harmonics,
oo N(q,m)
f(x) = f(rw) = L L rm fm.;(r) Ym,;(w) , (1) m=O i=1
where r is the length of x, r = lxl =(xi+ ... + x~) 1 12 , and wits direction, w = xfr E S9- 1 = {71 E /R9 : 1711 = 1}. The function Ym,i is a spherical harmonic of degree m. That is, the function x t--t lxlm Ym,;(x/lxl) is a homogeneous harmonic polynomial of degree m
on JR9. N(q,m) is the dimension of the vector space of all spherical harmonics of degree m on /Rq. The set {Ym,j : m E /N0 , j = 1, ... , N( q, m)} is an orthonormal basis in the Hilbert space L2(Sq-\ dO'q- 1 ) where O'q-1 is the surface measure of sq-t. Thus we have
rm fm,;(r) = j f(rw) Ym.;(w) d0'9-1(w) (2)
s•-t
and
oo N(q,m)
IIJIIL(R•) = L L llrm fm,;lli2(R+;r9-ldr) · (3) m=O j=1
4
Now an application of the Hecke-Bochner theorem (see Theorem 1.33) yields that the Fourier transform IF f is given by
oo N(q,m) (IF!) (rw) = L L (-i)m r"'(llm+q/2-dm.;) (r) Ym.;(w), r > 0, wE SH.
m=O .i=1
(4)
Here for 11 ~ -~, ll, denotes the Hankel transformation
00
(ll,g) (x) = j (xy)-" J,(xy) g(y) y2"+1 dy, x > 0, (5) 0
where J, is the Bessel function of the first kind and of order 11.
By means of the relations (1 ), (3) and ( 4), the characterization of Sa as given in Example 1 can be reformulated as follows: A function f on JR9 belongs to Sa if and only if there exist A, B > 0 such that
oo N(q,m) ~ ~ 11 k+m J 112 < B2A2kk2ka L L r m,j L,(R+;r•-t dr) - , k E DVo,
m=O j=l
and
oo N(q,m) I
.E .E llrl+m Hm+q/2-1 /m,jii~(R+;rHdr) < 00 ' lE DVo · m=O i=l
Similar characterizations follow for the s~aces SP and S~, see Chapter 2, Theorem 2.48. So for a function fin a Gel'fand-Shilov space, the corresponding fm.; and Hm+q/2-l fm.; are rapidly decreasing at infinity. It is shown that these properties are necessary and sufficient for the function fmJ to belong to a Gel'fa~d-Shilov space again (see Chapter 2, Theorem 2.21 ). Considerations of this type have led us to a study of the Hankel transformation ll11 on Seven• Sa,even• S~ven• S~,even• consisting of the even functions in S, Sa, SP, S~, respectively. Here it is understood that the definition (5) is supplemented by (ll,g) (x) = (ll,g) (-x) for x < 0, which makes Dl,g into an even function. Let us review some properties of the Hankel transformation. (i) For 11 ~ -~,the Laguerre functions
Jlf(x) = n · exp(-lx2 ) L"(x2 ) n E DV0 , ( 2f( + 1) )
112
n f(n+ll+1) 2 n '
constitute an orthonormal basis in the Hilbert space X 2,+i = L2(JR+; x2"+I dx). Here L~ denotes the generalized Laguerre polynomial,
e"' L:(x) = x-" 1 (dfdx)" [e-:r x'*'] , nE DVo.
n.
Now the Hankel transformation ll, can be extended to a unitary operator on X2,+t with the property ll;;1 = ll,, because
llv IL~ = (-1)" IL~, nE DVo.
5
The spaces Seven and S~,even with a ;::: j, can be characterized in terms of the expansion coefficients of their elements with respect to the basis {JL~ : n E JV0 }:
Let 11 ;::: -j. Then
Seven= {! E X2.,+1 : (nk(/, 1L:)x2v+1 ) E loo , k E lVo} ,
see Chapter 1, Theorem 1.8 and Chapter 2, Theorem 2.18. From these characterizations it follows that
lllv(Seven) =Seven , JH.,(s:,even) = s:, • ...,n •
(ii) For the remaining Gel'fand-Shilov spaces it is proved in this thesis that
see Chapter 2, Theorem 2.22. (iii) A function I E x211+1 can be extended to a function in S",everu if and only if there exist A, B > 0 such that
llxA: II!L2(R+J ::; BAkkka and llalll .. lli~<R+) < oo , k, I E lVo .
From the preceding property (ii) and Kashpirovskii's intersection result S!,even = S"·•ven n S~v•n• similar characterizations for the spaces S~ven and S~,eveu follow, see Chapter 2, Theorem 2.21. (iv) Products of Hankel transformations have the following properties:
2~-v+l oo
(llp(ll.,f)) (x) = r(v- p,) J f(y) (y2 x 2)"-p-l ydy, -~::; ft < 11'
"'
( 1 d )k
(llv+k(ll,J)) (x) = -; dx l(x) , 11;::: -~ , k E !No.
These relations suggested the idea of characterizing the spaces Seven• 8a,even• S~ven• S~,.ven• by means of the operators x and x-1 dfdx, cf. Chapter I, Corollary 1.21, and Chapter 2, Corollary 2.29. Returning to the expansion (1), we come to the conclusion that the basic function (rw)....,. r"' fm.;(r) Ym.;(w) belongs to S, Sa, Si3, S~, if and only if fm.i E Seven, Sa,even, S~ven• S!,even• see Chapter I, Theorem 1.37 and Chapter 2, Theorem 2.45. In particular, by taking m = 0, we obtain a characterization of the radially symmetric functions: A radially symmetric function I : J.R:1 -+ C belongs to S, S01 , Si3, 8~ if and only if there exists g E Seven, SQt,even• s:ven• S!,even• such that l(x) = g(lxl); see Chapter I, Theorem 1.36 and Chapter 2, Theorem 2.46.
The Radon transform of a function on /Rq is defined as the set of its integrals over the hyperplanes in J.R:l. A known result is the socalled projection tlworem (see Theorem 1.46), which provides the fundamental relationship between the Radon transformation 'R and the Fourier transformation IF,
6
('Rf) (p,w) = (27r)qf2- 1 j (Hi' f) (tw) e;pc dt , p E IR , wE sq-l . (6) R
Helgason [27] and Ludwig [37] characterized the image of the Schwartz space S under the Radon transformation 'Rand they presented several formulas for 'R-1• Since ffi'(S) = S, the characterization of 'R(S) comes down to the characterization of all functions
(p,w) ~--+ j f(tw) eipl dt , f E S . R
We give a new proof of the characterization of 'R(S), see Chapter 1, Theorem 1.47. The proof is based on the identification of s with a subspace of L2(/R X sq-1 ), see Chapter 1, Theorem 1.43. As a side result, we come to the solution of the following factorization problem: For which functions g and h does the function (rw) ~--+ g(r) h(w) belong to S? (see Chapter I, Theorem 1.44). In this thesis we start an investigation of the Radon transformation on the Gel'fand-Shilov spaces, see Section 2.6. We are mainly interested in the action of the Radon transformation on the basic functions (rw) ~--+ r 111 fm,i(r) YmJ(w) in the Gel'fand-Shilov spaces. We recaH that such a basic function belongs to Sa, Sf3, S~, if and only if fm.i E Sa,evet" S~ven• S~.even• respectively. Let us consider such a basic function,
(7)
By means of the projection theorem (6) and the Hecke-Bochner theorem (cf. (4)), we have
('Rf) (p,w) = (27r)q/2-1 (-i)m j tmflllm+q/2-Ifm,;) (t)eiptdtYmJ(w), R
p E IR , w E Sq-I . (8)
From the definition of the Hankel transformation in (5) with v -~,it follows that JIL 112
equals the Fourier-cosine transformation. Hence, the expression (8) can be written as
('RI) (p,w) = 9m,J(P) Ym,J(w), p E IR, wE Sq-t, (9)
with
9m,j(p) = (21r)(q-1112 (- ~).,. (DLt/2/Hmh/2-t!m,J) (p), pE IR. (10)
Due to the property (ii) of the Hankel transformation it follows that fmJ E Sa,even1 S~ven> S~,even• if and only if Urn,; E (d/dp)m (Sa,even), (d/dp)m (S~.,;,.,), (d/dp)m (S~,even)· For the evaluation of (10), we apply property (iv) of the Hankel transformation and the Rodrigues' formula for the Gegenbauer polynomial C:,{2- 1 , to obtain
00
9mJ(P) = A(q, m) j (r2 - p2)(q-3
)/2 C'/,{2
-1(p/r) rm /m,J(7') r· dr , p > 0 , (11) ,
where A(q, m) is a multiplicative constant. This formula can also be found in Ludwig [37J. Deans [7], [8] found the inversion of (ll), viz.
7
00
r"' /m,i(r) = ll(q,m)r2-q j (p2 -r2 )(q-3)12 C~2-1 (p/r)g~'j1l(p)dp, r > 0, (12) T
where 1-'(q, m) is a multiplicative constant. This led to our study of integral equations of Mellin-convolution type, on which there is a vast amount of literature. In the literature there are two common solution methods for this class of integral equations. The first method seems to be rather ad hoc: By clever guessing, sometimes supported by a formal application of the Mellin transformation, a solution of the integral equation is proposed. Then correctness of the proposed solution is shown by substitution into the integral equation, which requires the evaluation of a special convolution integral. The second method is based on fractional integral/differential operators. It is shown that the integral operator can be factorized as a. product of multiplication operators and fractional integral/differential operators. From this, the existence and uniqueness of the solution is inferred. However, in this approach solutions in a manageable form are not always available. In Chapter 3 we have worked out the latter method for integral equations of Mellinconvolution type where the convolutor contains a Gegenba.uer polynomial or more generally, a Gegenbauer function. After extending the Rodrigues' formula for Gegenbauer polynomials to Rodrigues-type formulas for Gegenbauer functions, we show that prod· ucts of fractional integral/differential operators can be reduced to integral operators with Gegenbauer function in their kernels, see Sections 3.4, 3.5. In Section 3.6 we show that the inverse of such an integral operator can be factorized as the product of a fractional differential operator and an integral operator involving again a Gegenbauer function. This result follows from an intertwining relation for fractional' integral/differential operators which in its simplest form reads
(d) 2n+l (1 d)n 2n+l(1 d)n+l - = - - x - - , n E /No . dx xdx xdx
To finish this introduction we briefly describe the contents of the separate chapters. Chapter 1 contains a detailed study of the Schwartz space S. In order to get a good understanding of the functions of q variables, much attention is first paid to the functions of one vari· able, see Sections 1.1, 1.2 and 1.3. In particular, the space Seveu(IR) is studied extensively in Section 1.2. In the analysis integral operations such as the Fourier transformation and the Hankel transformation play an important role, but also fractional integral/differential operators are involved. In Section 1.5 we give an introduction to the theory of spherical harmonics. Sections 1.4 and 1.6 provide results on the Schwartz space S( JR:l). Of special interest is the characterization of the radially symmetric functions and of the basic functions rm fmJ(r) YmJ(w), as elements of S(/Rq). Then the characterization of a general f E S(/R9 ) can be found by expanding f into a series with respect to spherical harmonics. Furthermore, in Section 1.6 the space S(/Rq) is embedded in L2(/R X sq-l) in such a way that the image of S(JR:l) under the Radon transformation R becomes obvious, cf. Section 1.7. The evaluation of the image R{rm /m,;(1') Y.n,J(w)} leads to special Gegenbauer transformations which are studied separately and in a broader context in Chapter 3.
The main objective of Chapter 2 is to carry over the results of Chapter l to the Gel'fandShilov spaces Sa, SP and S~. The set-up of Chapter 2 is very similar to that of Chapter l
8
on the Schwartz space. Because the Gel'fand~Shilov spaces are subspaces of the Schwartz space, we are able to strengthen and extend many of the results of Chapter 1. Only the factorization problem in SP and S! with p > 1, and the characterization of the images 'R(S .. ) and 'R(Se} with a> 0, are still open and require further investigation.
The plan of Chapter 3 is the following. In the first two sections we gather some relevant material on fractional calculus based on Weyl operators and on Riemann-Liouville operators. To avoid technical complications, we introduce the spaces s_ and s_, which consist of C""-functions on (0, oo ); the functions ins_ have a specified behaviour at oo and the functions in S._ have a specified behaviour at 0. All statements are in terms of these spaces. Section 3.3 is devoted to the derivation of Rodrigues-type formulas for Gegenbauer functions, which extend the usual Rodrigues' formula. for the Gegenbauer polynomials. In Sections 3.4 and 3.5 we introduce the so-called Gegenbauer transformations and in Section 3.6 we present several formulas for their inverses. Here we like to mention already that the inverse of a Gegenbauer transformation is again a Gegenbauer transformation. In Section 3.7 the theory of Gegenbauer transformations is utilized to solve certain integral equations of Mellin-convolution type. The solutions obtained are compared to results from the literature.
9
CHAPTER 1
Functions of rapid decrease
1.1 The Schwartz space S(JR)
In his treatise on the theory of distributions [45, Section VII.3] Schwartz introduced the space (S) of infinitely differentiable functions of rapid decrease on R'. In this section we gather some properties of this space for q = 1.
Definition 1.1 The Schwartz space S(JR) consists of all functions/ E C00(/R) for which
xk /(I) E Loo(IR) , k, lE /No .
The topology on S(JR) is defined by the countable set of seminorms
f ~---~> llxk f(I)IILoo(ll) , k, I E /No ·
This topology is metrizable and gives S( JR) the structure of a complete metric space.
The multiplication operator x and the differentiation operator d/ dx mapS( JR) continuously into itself. For f E S(JR) we have (I+ x2) f E L00(/R), whence we obtain S(JR) C Lp(JR), p;::: 1. In particular, we can define the Fourier transformation :F on S(JR) by
1 J . (:F/) (x) = m= f(y)e-'"' 11 dy, / E S(JR), x E JR. y21r ll
(1.1)
In [45, Section VII.6, Theorem XII], Schwartz has proved that the Fourier transformation :F maps S(JR) continuously onto itself. We give some characterizations of S(JR).
Theorem 1.2 Let f E C""(JR). The following assertions are equivalent:
(i) I E S(JR);
(ii) xk J(l) E L2(/R) , k, lE /No;
(iii) x" / E L2(/R) and Jl'l E L2(JR) , k, 1 E /No ;
(iv) x" / E Loo(IR) and f<'l E Loo(IR) , k, lE /No .
Proof. We follow the scheme (i) => (ii) => (iii) => (iv) => (i). Suppose f E S(JR). Then x" J(l) E S(JR) C L2(JR), k, lE /N0 • So (i) => (ii). Obviously (ii) ::::} (iii). Suppose f satisfies assertion (iii). Then for k, lE IN and x E lR we estimate
X
lx" f(x)il = 2 Re J tk f(t) (t" f(t))' dt 0
:5 2k llx~< /IIL,(Rl llx"-1 /IIL>!Ill + 2 llx2" /IIL,(IiJ 11/'IIL.(RJ ,
10
11!
lll'l(x)l2 -II<IJ(oW = 2 Re j l!'l(t) l!'+t>(t)dt ~ 2 1111'liiL2(.Rl llf<'+tliiL2!.Rl · 0
This proves assertion (iv). Finally, suppose I satisfies assertion (iv). Then for k E IN and x E IR we have, through integration by parts,
"' lxk f'(x)ll = 2 Re j tk f'(t) (tk f'(t))' dt = 2 Re [x2k-t 1(3:) (k/'(x) + x/"(x))
0
"' -J t 2"-
2 l(t) (k(2k -1) f'(t) + 3kt f"(t) + t2 f'"(t)) dt] . 0
From this result we conclude that x" f' is bounded on JR. Now replace I by f', then the argument can be repeated and an induction procedure yields I E S( /R). D
Next we want to present some functional analytic characterizations of S(IR). For that purpose we introduce some terminology. Let T be a self-adjoint operator in a Hilbert space H. According to Nelson [41], a vector I E H is said to be a C00-Vector for T if I E D(T"), nE /N. The set of C""-vectors forT is called the C00-domain ofT and is denoted by D00(T):
00
D00(T) = n D(Tn). n=l
The topology on D00 (T) is defined by the,countable set of seminorms
I ~ urn IIIH , n E /No .
{1.2)
This topology is metrizable and gives D""(T) the structure of a complete metric space. For a unitary operator U on H we have
D00 (UTU*) = U(D""(T)). (1.3)
We introduce some notations: lF' denotes the Fourier transformation on £2( /R), i.e. lF' is the unique unitary operator on £2(/R) such that lF'I = Fl, I E S(JR). P denotes the differentiation operator in £ 2(/R) defined on the domain
D(P) = {f E £2(/R): I is absolutely continuous, f' E £2(/R)}
by Pf if'. Q denotes the multiplication operator in £ 2(/R) defined on the domain
D(Q) = {f E L2(Dl): xf E £2(/R)}
by Ql xl. The operators /F', P and Q are discussed at length in Helmberg [29, Sections 16, 18, 19]. We quote some useful results. For f E £1(/R) n L2(B.) we have for almost all x E IR,
(JF'f) (x) = . ~ j l(y) e-ixy dy , (/F'* f) (x) = ~ j l(y) e'"'Y dy . (1.4) v2~ .R v2~ .R
11
The operators P and Q are unbounded and self-adjoint. For f E S(R) we have by interchanging differentiation and integration, P/Ff = IFQJ and PIF*f = -/F*QJ. Even more is true,
P = JFQJF*, Q = -IFP/F*. (1.5)
Thus it follows by means of (1.3) that
(1.6)
From Theorem 1.2 we obtain the following functional analytic characterization of S(IR):
Theorem 1.3
The topology on S(Hl) equals the intersection topology on D00 (Q) n D00 (P).
The operators IF, P and Q map S(Hl) continuously into itself.
Theorem 1.4
(i) IF(S(Hl)) = S(Hl) ,
(ii) Q"(S(Hl)) = {f E S(Hl): J!kl(O) = 0, k = 0, ... , n- 1} , nE IN,
(iii) P"(S(R)) = {f E S(R): j J(y)ykdy = 0, k = O, ... ,n-1}, nE IN. R
Proof. (i) From (1.6) and Theorem 1.3 we have
JF(S(R)) = JF(D00(Q)) n JF(D00(P)) = D00(P) n D00 (Q) = S'(R).
(ii) Let nE IN and let f E S'(R) with J(kl(O) = 0, k = 0, ... , n -1. We define the function g on m by
1 1
g(x) = (n _1)! j (1-tt-1 j(nl(xt)dt, x E 1R.
0
Then f( x) = x" g( x ), x E JR, and for k, l E /N0 and x E lR we estimate
lxk g(x)l :5 sup lu(e)l +sup lek+n oWl :5 ~ IIJ1")11Loo(R) + llxk !IILoo(R) , lel<1 lel>1 n.
lg(ll(x)l = 1 I ]1
(1- t)"-1 t1 J(n+ll(xt) dtl :5 _l_! - IIJ1"+I)IILoo(R) · (n-1)! (n+l)!
0
So g E S'(JR) by Theorem 1.2, and therefore, f E Q"(S(IR)). Conversely, suppose f E Q"(S(IR)). Then obviously f E S(fff) with J!kl(O) = 0, k = 0, ... ,n- 1. (iii) Observe that for f E S(IR) and k E /No,
12
(-l)"'(JF*Q[c f) (0) = <-1t J J(y)y[cdy. v21f B
Now the result to be proved follows from (ii) and the fact that for n E JN,
P"(S(/R)) JFQ"(S(/R)) = {J ES(/R): lF* J E Q"(S(/R))} .
The space S( JR) can be characterized by means of the Fourier transformation:
Theorem 1.5 Let f E L2(/R). The following assertions are equivalent:
(i) I E S(/R) ;
(ii) xk f E L00 (/R) and x11Ff E Loo(IR), k,l E lNo;
(iii) xk J E L2(/R) and x11FJ E L2(/R), k,l E JN0 •
Proof. We follow the scheme (i) => (ii) => (iii) => (i).
0
Since JF(S(/R)) = S(JR), (i) => (ii). Obviously (ii) => (iii). By use of (1.6) and Theorem 1.3, (iii) ::::} (i). 0
The space S(IR) can be characterized in terms ~f the Hermite expansion coefficients of its elements. The Hermite functions,
(1.7)
constitute an orthonormal basis in L2(/R). Here H,. denotes the Hermite polynomial,
H,.(x) (-1)" exp(x2) (djdxt exp(-x2
), nE JN0 ,
see Magnus, Oberhettinger and Soni [38, pp. 249, 252]. By means of [38, pp. 252, 254] we obtain the relations
Q,P,. J(n + I)/21/J,.+t + FJ2 '1/Jn-1 , nE lN,
PfjJ,. = -i J(n + l)/2f/J,.+1 + i Ffit/Jn-1 , nE lN,
(1.8a)
(1.8b)
(1.9)
For later use we need an L00-estimate for xk,p~l(x). From [18, 10.18(19)] we quote the inequality
11/J,.(x )I < 1.086435 'IT-1
/4 < 1 , x E JR ,
established by Cramer and Charlier. Next it can be shown by an induction argument based on ( l.Sa.,b) that
(1.10)
It also follows from (I.Sa,b) that the Hermite functions constitute an orthonormal basis of eigenfunctions of the operator P2 + Q2
,
13
(P2 + Q2)t/J.,. = (2n + l)t/J.,., nE /No· (1.11)
Now we arrive at the announced characterization of the space S(ll):
Theorem 1.6
(i) S(ll) = D""(Q) n D""(P) = D""(P2 + Q2)
= {! E L2(ll) : (n"(J, t/J.,.)L~(Ii)) E loo , k E /No} .
The equalities hold as topological vector spaces.
(ii) For f E S(JR) we have
00
f(x) = L (!, tPn)L2(1i) t/J.,.(x) , x E lR, n=O
where the series converges in S(JR). The series converges absolutely with respect to the Loo(IR)-norm and the series converges uniformly on JR.
Proof. (i) The first equality has been established in Theorem 1.3. Since the operators P and Q map S(JR) continuously into itself, we have S(JR) C D00(P2 + Q2
). We shall show that
Let f E D""(P2 + Q2). By applying (1.11) we derive fork E /N0 ,
00 00
sup Jnlc(J, tPn)L2(R)I2 S E Ink(!, tPn)L2(1i)l2 $ E J(2n +I )k (f, tPn)L2(1i)l2
~~ ~ ~
00
= L I((P2 + Q2)k J, tPn)L2(RJI 2 II(P2 + Q2
)k /JJL(Il) < 00 · n=O
Next suppose J E L2( JR) such that ( n"(J, tPn)L,(Il)) E loo , k E /No. Let k E /No. By (1.8a) there exist coefficients aj,~o(n ), j = 0, ... , k, such that
k
xkt/J,.(x)::;::: L aj,k(n) tPn+k-2i(x), nE /No, j=O
and, by (1.8a), it can be proved that
Jaj,k(n)JZ $ 2k(n + k)", j = 0, .. . ,k, nE /No.
Using the latter two results we deduce
00 00 k
L lj xk J(x)t/J,.(x)dxl 2 = L IL aj,k(n) (J,tP,.+k-2j)L2 (1~)1 2 n=O R. n:O j=O
14
k 00
:$ 2"(L sup J(n + 1) (n + k)"/2 (!, lPn+k-2j)L2 (.R)I)2 L (n + l)-2 < oo.
j=O nENo n=O
Hence J E D""(Q). By (1.9) it follows that (n"(IFJ, t,b,.)L2 (.Rj) E loo , k E /N0. So, by replacing f by /Ff in the above argument, we infer /Ff E D00 (Q). That is, f E D00(P), by (1.6). Now it is not difficult to see that the equalities hold as topological vector spaces.
(ii) Let f E S(JR) and let fN "L,~.;l (!, lPn)L2 (.R) t/Jn· Then f- fN E S(.R). We show that f- fN-+ 0 (N-+ oo) in D00(P2 + Q2). For leE IN,
00
11( P2 + Q2)" (J- fN )IIL(.R) L l(2n + l )" (J, Wn)~(R)I2
-+ 0 (N-+ 00) , n=N
by assertion (i). That is f- fN-+ 0 (N-+ oo) in D00(P2 + Q2) = S(R).
The remaining part of assertion (ii) follows by means. of the estimate Jt,b,.(x)l :$ l from {l.lO). D
Theorem 1.6 can also be found in Simon [46, Theorem lJ and in Jones [32, Section 5.2]. A more general result has been proved by Goodman (23, Section 6].
1.2 The subspace Seven(JR) and the1 Hankel transformation Hv
In this section we consider the subspace Seven(R) of even functions in S(.R). The topology on Seven(R) is the induced topology. Likewise, Sodd(.R) is the subspace of odd functions in S( IR). By use of Laguerre expansions and properties of the Hankel transformation we come to the main result: An even function f E L2(R) belongs to..Seven(R) if and only if both f and its Hankel transform Dl., f decrease at +oo more rapidly than any negative power of x {Theorem 1.12). Furthermore, the operators DI,DI.,, f.', v;::: -~,yield a fractional calculus for the operator -x-1 d/dx on Seven(IR.). We start with a straightforward consequence of Theorem 1.4.
Theorem 1.6a Let n E IN. Then
(i) /F(Seven(/R)) = Seven(R),
(ii} Q(Seven(/R)) = Sodd(.Jl) ,
Q2n{Seven(IR.)) = {! E Seven(R) : f<2"l(O) = 0, k = 0, ... , n- 1} ,
Q2"+l(Seven(R)) = {! E Sodd(.R) : j<2H 1l(O) = 0, k = 0, ... , n -1} ,
(iii) P(Seven(IR.)) = Sodd(.R),
P 2"(Seven(R)) = {! E Seven(R) : j f(y) y2k dy = 0 , k = 0, ... , n- 1} ,
R
P 2"+1 (Seven(R))={fESodd(/R): j f(y)y2"+1 dy=O, k O, ... ,n-1}. R
15
On the basis of Theorem 1.6 the space Seven(/R) can be characterized in terms of the Laguerre expansion coefficients of its elements. For v ;::: -!, the Laguerre functions
" 2r n + 1 1 2 " 2 (
( )
)
1/2
JL,.(x) = r(n +V+ 1} exp(-'ix) L,.(x)' nE lVo' (1.12)
constitute an orthonormal basis in the Hilbert space X2.,+1 = L2(JR+; y2"+1 dy). Here L~ denotes the generalized Laguerre polynomial,
ez L~(x) = x-" n! (d/dx)" [e-"' xn+v] , nE 8V0 ,
see [38, p. 241].
Lemma 1. 7 Let 11, p. ;::: -! with v f. p.. Then n
(i) JL~ = E A:;t,',.lL!:,, nE /No, m=O
where A"•"' is the upper triangular matrix with entries
A"·"'-- (r(n+l) r(m+p.+l))112
(v-p.)n-m m,n r( n + V + 1) f( m + 1) f( n - m + 1) '
0$m$n.
(ii) There exist constants C,,P > 0 and l.,,p E IN such that
l A"•~' I < C n1"·" l <_ m <_ n . m,n - II•J.I. ,
Proof. (i) The assertion follows from the formula [38, p. 249]
Lv - ~ (11 P,)n-m L~' mt n- L...J ( - )I m ' nE .IIYO •
m=O n m.
(ii) From Van Eijndhoven and De Gra.af [14, Lemma a.ll] we quote: For c,d > 0 there exists a constant Kc,d. > 0 such that
f(m+c)/f(m+d) S Kc,d.mc-d, mE IN.
By means of this inequality and by observing that I( v - p. ),._,.I :5 (I v - p.l),._m r(n- m+ lv- p.l) / f(lv- p.l), we estimate
IA~:~I 2 = r(n + p. + 1) /f(n + /) + 1) s Kll+l,v+l n"-v' nE IN'
IA:;;~ .. I2 :5 Kt,•+t K,.+u K~_,.p r-2(111 pi) n-" m"(n- m)21"-"H , I s m< n .
Now the assertion is readily obtained. D
Theorem 1.8 Let 11 :2: -!. Then
(i) Beven(/R) = {f E X2v+l : (n"(f, 1L~)X2v+ 1 ) E loo , k E /No} .
(ii) For f E Seven(/R) we have
"" f(x} = E (!, 1L~)X2v+ 1 JL~(x) , x E JR.+ , n=O
16
where the series converges in SeVfJD(/R). The series converges absolutely with respect to the Loo(JR+)-norm and the series converges uniformly on JR+.
Proof. The Hermite polynomial H2.,. can be expressed in terms of the Laguerre polynomial L;112 , [38, p. 240],
H2.,.(x) = (-1)"'22"n! L;;112(x2), nE /No.
Hence the Hermite function t/>2., can be expressed in terms of the Laguerre function JL;;112,
tP2n(x) = (-:;r IL;;112(x), nE /No.
Thus, for v = -~ assertion (i) is a corollary of Theorem 1.6. We next introduce the notation
Vv = {! E X2v+l : (n"(f, JL~)x2.,+I) E loo, k E llVo} •
(1.13)
Let v, p ~ -~ with v -:/: p. We show that Vv C Y~'. Let / E Vv and let a.,. = (!, IL~)x2.,+,. According to Lemma 1.7(ii) there exist constants Cv,~> > 0 and lv,~> E JlV, such that for k,m E IN,
(1.14)
Hence, the double series 00 00
" " a A"·" Hi' ~ L...J n m.,n m m=O n=m
converges absolutely in X2p.+l· Therefore, the double series converges in X2"+I with
00 00 oo n oo
2: L an A:;;~,. ILi,. " a " Av;~' /If = " a /Lv = f L,.; n L.-t m,n m L...J n n • m=O n=m n=O m=O n=O
Thus we have 00
f E x21'+1 with (!, 1Li,.)x2,.+1 = L a,. A:;;~.,. . (1.15) n=m
By using (1.14) we infer f E V". Now the embedding Beven(IR) = V_1 c V" c V_1 Seven(IR) proves assertion (i).
2 2
(ii) Let f E Seven(IR), let a.,. = (!, IL~)x2v+, and let !N E~;;01 a.,. IL~. We show that f- fN-+ 0 (N-+ oo) in Seven(IR). By using (1.11), (1.13) and (1.15) with f replaced by f- fN and p by -~,we have fork E JlV,
{(P2 + Q2)" (!- fN), tP2m)L,(R) = (4m + 1)" (!- JN, tPzm)L.(R)
= (-I) m y'2 ( 4m + l)k (/ - fN, fL;;.112)L2(R+)
00
E a All,-1/2 n m.n •
n=max(m,N)
17
According to Lemma. 1.7 (ii) there exist constants C = Cv,-l/2 > 0 and l = lv,-t/2 E IN, such that for k E IN,
iii(P2 + Q2)" (f- /N)IIL(R)
N oo oo oo
= E ( 4m + 1 )211 I E a,. A~:;.t/2 1 2 + E ( 4m + 1 )2
k I E a,. A~~ .. *l2
m=O n=N m=N+l n=m
N oo oo oo
:5C2 E <E la,.l(4n+1).\:n1)2 +C2 E m-2 (E la,.l(4n+l)11 n1+1)2
m=O n=N m=N+l n=m
00
:5 (C sup (la,.l (4n +I)" nl+3) E n-2
)2
nEN n=N
Now, by use of assertion (i), it follows that II(P2 + Q2)k (f /N)IIr,(R) - 0 (N ---+
oo), k E IN. That is f- fN---+ 0 (N---+ oo) in Seven(/R). Ffom Lemma 1.7 with p =-!,relation (1.13) and the estimate I.,P .. (x)j :51 from (1.10), it follows that there exist k,, l., E IN such that for x E JR+ and n E IN,
IL:(x)l :5 k,n1".
The remaining part of assertion (ii) follows by means of this estimate. 0
Let us introduce the Ha.nkel transformation H 11 for v ~ -~on L1(JR+; y11+112 dy), defined by
00
(H,f) (x) j (xyr" Jv(xy) f(y)y2"+1dy, f E L1(JR+; y"+112 dy), x > 0, (1.16) 0
and (Hvf) (x) = (Hlvf) (-x) for X< 0. Here J, denotes the Bessel function of the first kind and of order v. Observe that the integral converges absolutely, due to the asymptotic behaviour of the Bessel function, J 11(t) = O{t") as t! 0 and J.,(t) = O(t-112
) as t-oo. Since (111't)112 J_l(t) =cost, the Hankel transformation H_l equals the Fourier-cosine transformation,
2
2
(HI_~!) {x) /! j f(y) cos(xy)dy, f E Lt(JR+), x > 0. 0
From [38, p. 244J we quote the formula
00
L:(x) = i(-Itx-v/2ex/2 j e-y/2yv/2 L:(y) Jv(..fi'ii)dy. 0
By replacing x by x2 and y by y2 it follows that
DfviL:=(-1)"/L:, nE !No.
(l.l7)
(1.18)
18
So Df., can be extended to a unitary operator on X2.,+1, 00
Df.,J = E (-It (J,JL~)x2•+1 /L~, f E x2.,+1 , n=O
with the property
Df~ =I.
(1.19)
(1.20)
Theorem 1.9 Let v 2::: -~. Then the Hankel transformation Df., maps Seven(IR) continuously onto itself.
Proof. The theorem is a consequence of (1.18) and Theorem 1.8. 0
In order to characterize Seven(IR) by means of the Hankel transformation Df., we present two auxiliary results. These results are interesting in themselves and will be applied once again in Section 2.2.
Lemma 1.10 Let-~$ p. < v, let a> v+ 1 and let f E L2 (JR+; (1 +y2)" y2"+1 dy). Then Df.,j E X2,.+1 and for X > 0,
2"-v+l oo
(Df .. (Df .. J)) (x) = r( ) j f(y) (y2 - ±2)"-,.-1 y dy . ~ • v-p.
X
Proof. For convenience we write X~.,+t = L2(JR+; (1 +y2)" y2"+1 dy). The proof is divided into three steps. Step 1. By means of the Cauchy-Schwarz inequality we have
00 00 00 J 1/(y)l y"+l/2 dy $ (j lf(YW (1 + y2)" Y2v+l dy)1/2 (j (1 + Y2t" dy)l/2. 0 0 0
So f E L 1 (JR+; y"+l/2dy). Now, by use of (1.18) and the inequality (cf. Watson [49, 3.31 ( 1)])
IJ.,(z)l $ lz/21" exp(l lm zl) / f(v + 1), z E C, (1.21)
we estimate for x > 0,
00 00
I(Df.,J) (x)l $ j l(xy)-" J.,(xy) f(y)y2H
1Idy $sup le-" J.,(OI j lf(y)IY2"+ldy 0 (>O- 0
< 2-v 11!11 Q (100
Y2"+1(1 + y2)-" dy) 112
- r(v + 1) x2•+1 0
(2-2v-1 r(a _V _ 1))1/2
= r(a) r(v + 1) ll!llx;.+, .
Hence
19
Since Ill., is a unitary operator on X2.,+t and since x:.,+t c X2 ... +t, we have
00 00
j I(IH11f) (x)j2 x2"+1 dx :5 j I(IH,J) (x)j2 x2
"+l dx :5 11/11~2~+• :5 11/11~;.,+1 . 1 1
By combining the latter two results it follows that IH11 is a bounded operator from X211+t into X2,.+t·
Step 2. By means of the Cauchy-Schwarz inequality we have
00 00
j I j f(y) (y2- xzy·-~t-1 ydyj2 x2"+t dx 0 X
00 00
:5 j {j l/(y)l2 (y2 _ x2)v-p-l (1 + y2)a Y dy 0 :r:
\ 00
j (y2- xz)"-~<-1 (1 + y2)-"' y dy} x2~'+1 dx .
"' By substituting y2 - x 2 ( x 2 + 1) t we obtain for x > 0,
00 00
j (y2- xzy-~<-1 (1 + Y2t"' ydy = !(x2 + qv-,.-a j t"-~<-1(1 + tt"" dt X 0
Moreover,
00 00
j {j l/(y)l2 (y2- x2)v-~<-1 (1 + y2)<> y dy} x21'+1 dx 0 X
00 y
= j l/(y)l2 (1 + y2)"' y{j (yz- xz)v-~<-1 x2~<+l (lx} dy 0 0
Let us introduce the operator Fj<,V on x~ ... +l' defined by
00
(F,., .. f) (x) j f(y) (y2- x2)"-~<-1 y dy, .1: > 0. X
20
Then it follows that F,.,v is a bounded operator from X2v+J into X2,.+1.
Step 3. First suppose v - p, > 1. Then the double integral
00 00
, j j (xtt"'J,.(xt) (ty)-v Jv(ty) J(y)y2v+lt2"'+1dydt 0 0
converges absolutely. Hence, by means of the integral formula (38, p. 100],
j J,.(at) Jv(bt) t"'-v+1 dt = oo { 21.1-v+l a"' b-v(b2 - a2 y-~.~- 1 [f(v- p,)]-1 , b >a ,
o 0, b<a,
we infer 00 00
( Dl,.( Dlvf)) (x) = J (xtt"' J,.( xt) { J ( ty rv Jv( ty) f(y) y2v+l dy} t 2"'+1 dt 0 0
00 00
= x-"' j {j J,.(xt) Jv(ty) t~>-v+l dt} J(y) yv+1 dy 0 0
(1.22)
Next suppose 0 < v- p, :S 1. Let g E Cgo(JR+), the space of C 00-functions op JR+ with compact support. Since (d/dx) [xv+l Jv+J(x)] = xv+l Jv(x), see [38, p. 67], we have through integration by parts
Then, by use of (1.22) with v replaced by v+ 1 and f replaced by (x- 1 dfdx)g, we deduce
(Dl,.(Dlvg)) (x) = -(Dl,.(Dlv+J((x-1 dfdx)g))) (x)
2~>-V 00 - J '( ) ( 2 2)V-IJ d 0 = r(v- Jl + 1) g y y -X y ' X> . :r
Integration by parts yields (Dl,.(Dlvg)) (x) = (F,.,vg) (x), x > 0. Since the space Cgo{JR+) is dense in X2v+1 and since both Dl,.Dlv and F,.,v are bounded operators from X2v+J into x2,.+t. the theorem follows. 0
In Section 3.1, formulas (3.7) and {3.8), we define the Erdelyi-Kober operators W~, A E JR, on the space S .... (JR+) = {! E C00 (JR+): xk j<1l E Loo([l,oo)), k,l E .DV0 } by
21-~ 00
(Wd) (x) = r{A) j J(y) (y2- x2 )~- 1 ydy, f E S .... (JR+), x > 0, A> 0,
:r .
(1.23)
21
Furthermore, we show that the operators JI.V~, A E JR, constitute a group on S-(fi+); cf. Theorem 3.6(i). By using this property and the relations (1.20), (1.23) and Lemma 1.10, it follows that
H,.Dl,f = JI.V,_,.f, f E Seven(/R), p.,v!;:: -~.
So, due to Theorem 1.9:
The operators JI.V~, A E JR, constitute a group of continuous operators on Seven(Dl).
In particular, the operator x-1 d/dx maps Seven( m.) continuously onto itself.
(1.24)
(1.25)
Lemma 1.11 Let -i 5 p. < 11 and let W be a nonnegative function on JR:*-, such that (1 + x2 ) 11_"_1 W(x) is nondecreasing on [b, oo) for some b > 0. Let f E X2.,+1 such that
xk f E L2(JR+) and x 1 H,J E Lz(JR+) , k, I E /No .
Suppose in addition that x"-!J W(x) llvf E L2(m.+). Then
(1 + x 2) 11-v-l W(x) H,.f E Lz(JR+) ·
Proof. Since xk f E L2( m.+), k E IV0, it follows that f E L1 (m.+; y!J+I/'l dy ). So H ,.! is continuous by (1.16). Therefore, (1 + x2) 11-v-t W(x) Hp./ E L2 ([0, b]). Since H'; = I and since H,f E L2(JR+; (1 + y2)"+2 y2"+l dy) we have by Lemma 1.10 with a= v + 2 and f replaced by llvf,
r:<:-:f> j 1(1 + x 2 )p.-v-l W(x) (Of,..!) (xW dx b
"" "" 5 j <J (I+ y2)~J-v-1 W(y) I(Dfvf) (y)l (y2- x'ly-p.-1 y dy)2 dx b .,
"" 00
5 j <J IW(y) (Of.,!) (y)l2 (yz- xz)v-~<-l dy)
6 "'
"" <J (1 + y2)2p.-2v-3/2(y2 _ xz)"-~<-1 ydy)dx
"' tB(v- p., 11- p. + 3/2)
"" "" j (j IW(y) (Dfvf) (yW(Y2- x2)"-~<-1 dy) (l + x2)~<-v-3/2dx b "'
= iB(v- p., 11- p. + 3/2)
00 " j <J (yz xz)v-,..-1 (1 +xz),..-v-3/Zdx) IW(y) (Dl,J) (yWdy. b b
22
Since the inner integral admits the estimate
V 1 ~ j (y2 _ x2)v-11-1 (l + x2)~~-v-3/2 dx $; b j (y2 _ x2y-~~-1 x dx
6 b
"2
2"-21' < _!_ J ( 2 _ )V-j.<-l d _ -:-:!f'-;----;-- 2b y X X - 2b(v -p) '
0
we obtain the wanted result because of the assumption x"-~-< W(x) Dl.,f E L2(JR+). 0
We now come to the main result of this section.
Theorem 1.12 Let v ~ -~ and let f E X2v+l· The following assertions are equivalent:
(i) f E Seven(IR.) j
(ii) xkJ E L00(JR+) and x 1 Dl.,j E Loo(JR+), k,l E /No;
(iii) x~<J E L2(m:+) and x1 DI,J E L:~(JR.+) , k, lE /NQ .
Proof. We follow the scheme (i) => (ii) => (iii) => (i). Since DI,-(Seven(IR.)) = Seven(IR.), (i) => (ii). Obviously (ii) => (iii}. Suppose f satisfies assertion (iii). If v =-~,then f E Sewm(IR.) by (1.17) an:d Theorem 1.5. Next suppose v > -!· For I E /N0 we introduce the function W(x) = (1 + x2)1+v+3/2•
Then the functions J and W satisfy the assumptions of Lemma 1.11 with I' = -~. Therefore, (1 + x2) 1 Dl_tf E L2(JR.+). So f E Seven( JR.) as in the case v = -!· 0
As a. consequence we obtain a. functional analytic characterization of Seven( JR.).
Corollary 1.13 Let v ~ -1/2. Then
(i) Seven( JR.)= D00(Q2) n D00(DivQ2 DI,),
where Q is the self-adjoint operator defined on {J E X:~ .. +t : xf E X2v+d by QJ = xf.
Proof. (i) Let J E X2v+t· From (1.3) with T replaced by Q2 and U by Dlv it follows that f E D""(DI.,Q2 Dl .. ) if and only if DI,J E D""(Q2
). So we have to show that the following assertions are equiva.lent.
(1) f E Seven(IR.) i
(2) x" j E X2.,+1 and x1 ll.,j E X2.,+1 , k, l E /No .
23
Suppose I satisfies (2). Then f E L1(.fi+; y"+l/2 dy) and so IH.,f is continuous by (1.16). Similarly, IH.,f E L1(.fi+; y"+l/2 dy) and so f = JH.,(JH,f) is continuous by (1.16). Hence f satisfies assertion (iii) of Theorem 1.12 and therefore, f E Seven( B.). The converse, (I)'* (2), follows from the fact that JH.,(Seven(.R)) =Seven( B.).
(ii) F:rom [38, pp. 241, 243] we deduce
x2 L~ = -((n + 1) (n + v + 1))112 L:+l + (2n + v + 1) L:- (n(n + v))l/2 L:_1 ,
(~/dx2 + (2v + l)x-1 d/dx + (4n + 2v + 2- x2)) L: = 0, nE JN0 •
By means of these relations and {1.18) the assertion follows. D
1.3 The space S(JR+) and the Hankel-Clifford transformation 'H11
In this section we introduce S(.fi+), the space of rapidly decreasing functions on .fi+. We present the connection with each of the spaces S(.R) and Seveu(.R). Because of these connections we are able to establish characterizations of S(JR+) which are similar .to those of S(.R) and Seven(.R) as given in Sections 1.1 and 1.2. Furthermore, it follows that Seven(lR) can be characterized by means of the operators x and x-1 d/dx (see Corollaries 1.16 and 1.21).
Definition 1.14 The space S(B.+) consists of all functions 9 E Coo( B.+) for which
xk9(l) E Loo(JR+), k,l E /No·
The topology on S(JR+) is defined by the countable set of seminorms
9 ~--+ llxk g(I)IILoo(R+) , k, lE /No ·
This topology is metrizable and gives S(JR+) the structure of a complete metric space.
We present the connection between S(JR+) and S(JR), and the connection between S(JR+) and Seven(JR):
Theorem 1.15
(i) S(JR+) = {g E C'=><>(JR+): g(x) = f(x) , x E JR+, for some f E S(.R)} ,
(ii) Seven(JR) = {! E coo(JR): f(x) = 9(x2)' X E JR, for some 9 E S(JR+)}.
Proof. (i) Let 9 E S(JR+). By means of Borel's theorem, see Zuily [52, Chapter 1, Exercise 1], there exists a function I E C00(JR) such that /(x) = g(x), X E JR+, and f(x) = 0, x :5-1. Clearly f E S(JR). The converse is obvious.
(ii) Let f E Seven( JR). Introduce the function 9 : JR+ --+ C by g( x) = f( .fii), x E JR+. By (1.25) the operator d/dx maps Seven(lR) onto itself. So for k,l E /N0 ,
sup ixk g<'>(x)i =sup lx2k g<1>(x2 )1 =sup lx2k(!x-1 d/ll.r:)1 J(x)l < oo . .:>0 .:>0 .:>0
24
Hence g E S(JR+) and /(z) = g(z2) , z E JR. The converse is obvious. 0
As a. straightforward consequence of Theorems 1.2 and 1.15 we obtain the following characterizations of S(JR+) and of Seven(.IR):
Corollary 1.16 Let g E C""(JR+). The following assertions are equivalent:
{i) g E S(JR+) ;
(iii) x"g E L2(JR+) and g<1l E L2(JR+) , k, lE /No;
(iv) xkg E L00(JR+) and g(l) E Loo(JR+), k,l E /No.
Let I E C00(JR+). The following assertions are equivalent:
(i) ! e S.,.,en(.IR) ;
(ii) x"(x-1 dfdx)1 f E Loo(JR+), k,l E /No;
(iii) x"(x-1 dfdx)1 f E L2(JR+), k,l E /No;
(iv) x"' f E L2(JR+) and (x-1 dfdx)1 f E Lz(JR+) , k, lE /No;
(v) x" f E Loo(JR+) and (x-1 d/dx)1 f E Loo(JR+) , 1
k, lE /No .
For 11 ;::: -i we define the Fourier-La.guerre functions£~, nE /N0 , by
C~(x)= (r(~(::;l)r/2
exp(-!x)L~(x), nEIN0 ,
and the Ha.nkel-Clifford transformation 1-lv on L1(JR+; y"l2-114 dy) by
00
(1-lvg) (x) = t J (xy)-vf2J.,(.jXY)g(y)y"dy' 9 E Lt(fR+; yvf2-1/4dy)' 0
(1.26)
X> 0. {1.27)
By means of the operator T,, defined by
(T,g) (x) = g(x") , x > 0 , p E IR , (1.28)
it follows from {1.12), (1.16), (1.26) and (1.27) that
C~ = ~ Ttt21L~ , n E /No , 7-l, = T112 HI., T2 . (1.29)
The Fourier-Laguerre functions£~, n E /N0 , constitute an orthonormal basis in the Hilbert space X.,= L2(JR+; y" dy). By use of (1.18) and {1.29) we have
1-t, £~ = ( -1 )" £~ ; n E 11/o •
So 'H., can be extended to a unitary operator on X,, 00
1-t.,g = L (-1)" (g,C~)x ... £~, g EX,, n=O
with the property
?{f,=l.
25
(1.30)
(1.31)
(1.32}
Due to Theorem 1.15(ii) the results for Sew.n(B.) established in Section l.2 can be translated into results for S(B.+). Let us gather these results.
Theorem 1.17 Let 11;::: -~. Then
(i) S(B.+) {J EX,: (nk(g,C~)xJ E loo, k E /No}.
(ii) ForgE S(B.+) we have
00
g(x) = L (g,C~)x., C~(x), x E B.+ , n=O
where the series converges in S(B.+). The series converges absolutely with respect to the L00(B.+)-norm and the series converges uniformly on JR+.
(iii) The Hankel-Clifford transformation 'H., maps S(B.+) continuously onto itself.
(iv) Let-~ :$ p, < 11, let o: > 11 + 1 and let g E L2(JR+; (I+ y)"' y" dy). Then 1-t.,g EX~' andforx > 0,
( 1.33)
In Section 3:1, formulas (3.3) and (3.5), we define the Weyl operators WA, >. E B., on the space K .. (B.+) = {g E C00(B.+): xk g(l) E Loo([l,oo)), k,l E /No} by
1 00
(WAg) (x) = r(>.) J g(y) (y- x)'~-l dy' g E S .... (JR+)' X> 0' >. > 0, .,
( 1.34)
Furthermore, we show in Theorem 3.3 that the operators W,,, >. E /R, constitute a group on S_.(B.+). By using this property and the relations (1.:32), (1.:33) and (1.34), it follows that
1t,..1t.,g = 2~"-" W.,_,..g, g E S(JR+), p,11 2: -~ .
So, due to Theorem 1.17(iii):
The operators WA, >. E /R, constitute a group of continuous operators on S(B.+).
(1.35)
(1.36)
26
In particular, the operator dfdx maps S(m.+) continuously onto itself. The space S( m.+) can be characterized by means of the Hankei-Clifford transformation 'H,:
Theorem 1.18 Let v 2:: -l and let g EX,. The following assertions are equivalent:
(i) . g E S(m.+);
(ii) xkg E L00(JR+) and x1'H,g E Loo(JR+) , k, lE /No;
(iii) xkg E L2(Dl+) and x1'H,g E L2(JR+), k,l E /No.
As a consequence we obtain a functional analytic characterization of S(m.+).
Corollary 1.19 Let v 2:: -~. Then
(i) S(JR+) = D00(Q) n D 00('H.,Q'H.,) ,
where Q is the self-adjoint operator defined on {g EX.,: xg E Xv} by Qg = xg.
(ii) ('H.,Q'H,) c~ = ( -(2..[i d/dx) 2- 2(2v + 1) d/dx) c:
( 4n + 2v + 2 - x) C~ , n E /No .
The subspace S .... (JR+) = {g E S(Df,+) : g!1>(o+) = 0 , l E /N0 } admits the following characterization.
Theorem. 1.19a Let v 2:: -l· Then a function g € S(JR+) belongs to S ... (JR+) if and only if
~( C") (f(n+v+l))t/2 k 0 kIN. t;;Q g, nX. f(n+l) n , E 0·
Proof. Let g E S(JR+). Since the operator dfdx is continuous on S(JR+) by (1.36), we have
00
(dfdx)k g(x) 2: (g,C:)x. (dfdx)k C:.(x), x E JR+ , k E /No, n=O
where the series converges in S(JR+), see Theorem l.l7(ii). Now, by using the properties [38, pp. 240, 241],
L:(o) f(n+v+l)/(r{n+l)f(v+l)), (dfdx)L:.(x)=-L:.!~(x), nEIN,
the function g belongs to S ... (JR+) if and only if
0 lim ((dfdx)k [e"'l2 g(x)]} x!O
00
= lim 2: (g,C:.)x. (dfdx)k [e"'/2 c:.(x)] x!O n=O
- ( -l)k 00 V
- f(v+k+l) E (g,.Cn)x, ( r( n + v + 1)) 112
r( n + I ) k E /No.
f(n+l) r(n-k+l)'
27
This proves the theorem because the latter series converges absolutely by Theorem 1.17(i) and because f( n + 1) / f( n - k + 1) can be expressed as a linear combination of ni , j = 1, ... ,k. 0
Theorem 1.19a. can also be found in Duran [10, Theorem 3.2]. At the end of this section we prove a characterization of S(JR+), inspired by Theorem 1.18 and Duran's paper [11]. Here is the idea: By using Theorem 1.17(iii), (1.35) and the fact that 'Ht is a. unitary operator on X,, we have forgE S(B.+) and lE /N0 ,
11:~:'12 'HoYIIL,(R+) = II'Ht 'Houllx, = 2'11:~:'12 g<'liiL,(R+) .
So by taking g E S(JR+), v = 0 and by replacing k by k/2 and l by l/2, Theorem 1.18(iii) admits the reformulation
x"l2g E L2(JR+) and x112g(l) E L2(B.+) , k, lE /No .
We show that these conditions on g E C""(JR+) imply g E S(B.+).
Theorem 1.20 Let g E C""(Dl+). The following assertions are equivalent:
(i) g E S(Dl+) ;
(ii) z(lc+l)/2g<1l E Loo(m.+) , k, 1 E /No;
(iii) x<~<+ll/2g<1l E L2(JR+), k,l E !No i
(iv) xkf2g E L2(m.+) and x112g(1l E L2(B.+), k,l E /N0 ;
(v) x,kf'lg E L00 (Dl+) and x112g(l) E Loo( m.+) , k, lE /No .
Proof. We follow the scheme (i) => (iii) =? (iv) =? (i) =? (ii) =? (v) => (iv). The implications (i) =? (iii) :::} (iv) and (i) =? (ii) =? (v) need no explanation. Suppose g satisfies assertion (iv). Introduce the function f by f(x) = g(x +I), x > 0. Then for x > 0,
( 1.37)
By using (1.37) with I 0 or k: = 0, and Corollary 1.16, it follows that f E S(m.+). For l E /N0 we have
1 I I
<J lu<IJ(v)ly'l2-lf4dy)2 :5 <J lu('l(yWy'dy) <J Y-li2(Ly) < oo. 0 0 0
Since f E S(m.+), we infer g<1l E L1(Dl+; y112- 114 dy). By (1.27), 'H,g<1l is continuous with
00
('H,,g(l)) (x) = ~ j (xy)-112 J,(yxy) g<'l(y) y' dy ' x > 0. {1.38) 0
Next we want to apply integration by parts. For that purpost~, we first show that
28
(1.39)
For 0 < x < 1 and l E .IN we have
1 l = 12 Re J y'/2 g(l-ll(y) [2 yl/2-lg(l-l)(y) + yl/2g(l)(y)] dyl
"' 1 l
~ 2 J ( 2 yl-1 I g(l-1)(y)l2 + IY(I-1)/2 gll-ll(y)y'l2g(l)(y)l) dy .. Cl() Cl() ""
~ l j y'-1 lu''-•>(y)l2 dy + 2(j Yt-1 lu<'-•>(y )12 dy )t/2 <J Y'lu<'>(y )12 dy)1/2 . 0 0 ; 0
Hence x112g(l-1) E L00([0, 1]). Since f E S(JR+), we infer x 112gll-t) E L00 (JR+). By using (1.21), (1.39) and the fact that f E S(JR+), we have for x > 0 and I E .IN,
(xy)-112 J1(vfxii)g(l-1l(y)y1-o for y.J.O andfor y-oo.
Hence by means of the relation (d/dz) [z" J.,(z)] = z" J,.:. 1(z), see [38, p. 67], we obtain through integration by parts in (1.38),
00
('Hzu<'l) (x) = j (xyr<t-tJ/2 Jz-t(vfxii) glt-1l(y)y'-•dy = -"-i('Hz-19<~-1)) (x). 0
By repeating this procedure we have
'Jt,g<'l = ( -2)-1 'Hog , lE ./No .
Now, since 'Ht is a unitary operator on X, we deduce for lE ./N0 ,
00 00 00
j x11(1tog) (x)j 2 dx = 221 j x1I{'Hzg11l) (xWdx = 221 j x1lg<l}(xWdx < oo. 0 0 0
Hence g E S(JR+), due to Theorem 1.18. Finally, suppose g satisfies assertion (v). Introduce the function f by f(x) = g(x + 1), x > 0. By use of (1.37), with k = 0 or l = 0, and Corollary 1.16 it follows that f E S(JR+), as before. Since x112g(l) E L00(JR+), lE .IN0 , and f E S(JR+), we estimate for k,,l E .IN0 ,
00
j xk lu(xW dx ~ sup jg(xW + j xk lf(x- lW dx < oo , R+ O~:t~1 1
00
j x1 lg<1>(x)l2 dx::S: sup lx112gl'l(xW+ j x1 lf(tl(x-IWdx<oo. R+ O~x$1 1
29
That is, g satisfies assertion (iv). 0
As a straightforward corollary of Theorem 1.15(ii) and Theorem 1.20 we obtain the following characterization of Seven(/R):
Corollary 1.21 Let f e C00(JR+). The following assertions are equivalent:
(i) / E Seven(/R);
(ii) x"+1(x-l dfdx)1 / E Loo(JR+), k,l E /No;
(iii) x"+'(x-1 dfdx)1 f E L2(JR+) , k, lE /No ;
(iv) x" f E L2(JR+) and x1(x-1 dfdx)1 / E L2(JR+) , k, lE /No ;
(v) x" / E L00(JR+) and x1(x-1 dfdx)1 / E L00(JR+) , k, lE /No .
1.4 The Schwartz space S(JR:l)
The Schwartz space S(JR'I) is the space of infinitely differentiable functions of rapid decrease on /Rq. In this section we gather some properties of this space for q ~ 2.
Definition 1.22 The Schwartz space S(/Rq) consists of all functions f E 0 00(/Rq) for which
Here we use the multi-index notation x" = x~1 • •• • ·x~q and {)1 8~ 1
• ••• ·fJ!q 'With Oj = fJjfJxj, j 1, ... , q. The topology on S(JR'I) is defined by the countable set of seminorms
/ 1-+ llx"d /IILoo(R9) , k, l E IN6 . This topology is metrizable and gives S(/Rq) the structure of a complete metric space.
The multiplication operators X j and the differentation operators aj , j = 1, ... , q, map S(JR'I) continuously into itself. For f E S( JR'I) we have (I+ lxl 2 )q f = (I +xi+ ... +x;)q f E L00(/Rq), whence we obtain S(JR'I) C £11(/R'I), p;::: I. In particular, we can define the partial Fourier transformations :F; , j = 1, ... , q, and the Fourier transformation :F on S( /Rq) as follows. For f E S(JR.q) and x E JR'I,
(1.40)
(:Ff) (x) = ((:Ft ... :Fq)J) (x) = (211'tqf2 j f(y)e-i(x·JJldy. ( 1.41) 1'!9
30
In !45, Section VII.6, Theorem XII], Schwartz has proved that the (partial) Fourier transformation( s) :F, :F1 , ••• , :F, map S( JR:l) continuously onto itself. Next we want to present some (functional analytic) characterizations of S(JR:l). For that purpose we introduce some terminology. Let T1 , ••• , T9 be mutually strongly commuting self-a.djoint operators in a Hilbert spa.ce H. Then we write
T (T~>···•Tq),
ITI = (T. T)1'2
'
T k - T"l T"q k E IAT'l - 1 • • • q , ~•o '
BTO (BT10, ... ,BT9C), B, 0 operators in H.
The joint 0""-domain ofT, denoted by D""(T), is defined by \
D""(T) n D(T").
The topology on D""(T) is defined by the countable set of seminorms
f r--t IIT"JIIH' k E INJ.
(1.42)
This topology is metrizable and gives D""(T) the structure of a complete metric spa.ce. For a unitary operator U on H we have
(1.43)
We introduce some notations: Let j E { 1, ... , q}. IFj denotes the j-th partial Fourier transformation on L2 ( IRq), i.e. IF; is the unique unitary operator on L2 (/R9 ) such that IFj/ :Fjf , f E S(/Rq). IF denotes the Fourier transformation on L2(/Rq), i.e. IF= IF1 ••• IFq. Pi denotes the partial differentiation operator in L2(1Rq) defined on the domain
D(Pi) = {! E L2(/R9): f is absolutely continuous with respect to
by Pj/ = iOj/. Q; denotes the multiplication operator in L2 (/Rq) defined on the domain
D(QJ) {! E L2(/R9 ) : xif E L2(JR:l)}
by Qj/ xj/. For f E L1(/R9 ) n L2 (/R9 ) we have for almost all x E /R9 ,
(IFJI) (x) I
(IF!) (x) = (211't9' 2 j J(y)e-i(.,·v)dy, (1.44) R,q
31
(F* f) (x) = (211')-q/2 j f(y) ei($·11) dy . (1.45) R9
The operators P; a.nd Q; a.re unbounded and self-a.djoint and they satisfy the relations
P; = lF;Q;IFj = IFQ;F* , P = FQF* , lP I = F IQI F* , (1.46)
Q; = -F;P;IFJ' = -FP;F*. Q = -FPF*. IQI = -FIPIF*. (1.47)
Thus it follows by means of (1.33) a.nd (1.43) that
D00(P;) = F;(D00(Q;)) = F(D00(Q;)), D00(P) = IF(D00(Q)),
D00(IPI) = F(D""(IQI)) ,
~(Q;) IF;(D00 (P;)) = IF(D00 (P;)) , D00(Q) = /F(D""(P)) ,
D""(jQI) = IF(D00(1PI)) ·
It is not difficult to show that q
D""(Q) = Doo(IQI) = n Doo(Q;). j=l
Hence, by use of (1.48), we also have
q
D00(P) = D00(jPI) = n D 00 (P;) . j=l
(1.48)
(1.49)
(1.50)
{1.51)
In the sequel 6 denotes the Lapla.cian: 6 -IPI2• We give some classical analytic char· a.cterizations of S(IR'~).
Theorem 1.23 Let I E C00(JR9). The following assertions a1·e equivalent:
(i) I E S(JR'~) i
(ii) x~<olf E L2(JR'~), k,l E IN3;
(iii) xk f E L2(JR9) and fJl I E L2(IR9) , k, lE INJ ;
(iv) lxlk/ E L2(1lq) and t::,,f E L2(JR9), k,l E /No;
(v) xU E L2(JR9) and o;1 E L2(1l'~) , j = 1, ... , q, k, lE /No ;
(vi) xkf E L00 (JR9) and 81/ E Loo(ll'~), k,l E INJ;
'(vii) xjf E L00 (JR9) and 011 E Loo(.IR9), j = 1, ... ,q, k,l E /No.
32
Proof. We follow the scheme (i) # (ii) # (iii) # (iv) # (v), (i) => (vi) => (vii) => (v}. The assertions (iii), (iv) and (v) are mutually equivalent by (1.50) and (1.51); the implications (ii) => {iii), (i) => (vi) => (vii) need no explanation. Thus it remains to be shown that (i) # (ii), {iii) => (ii) and (vii) => (v). Suppose f E S(ll"). Then xka' f E S(ll') C L2(1l'), k,l E BVJ. So (i) => (ii). Conversely, suppose f satisfies assertion (ii). For k,l E BVJ and x E ll' we have
(211")912 fxlc(Q'f) (x)l $ j I(F(xka'f)) (y)fdy li9
= J (F((I- a)• (:~NJ'f))) (y) (I+ IYI 2t' dy Ji9
$ 11(!- a)' (x"O' /)I[L,(Ii9) 11(1 + IYI2t"IIL,(Ii9) .
That is I E S(ll"). Next suppose f satisfies assertion (iii). For k, I E BV0 we have
j <J xilclf(xhyWdxt)dy < oo and j <J l(oU)(x~>y)f2 dxt)dy<oo. Jiq-1 R Jiq-1 li
So for almost all y E ll'-1,
j x~" 1/(xt. y)l 2 dx1 < oo and j 1(8!1) (x~oYW dx1 < oo , k, lE lVo . R R
That is, according to Theorem 1.2:
The function x1 ~----+ f(xt.y) E S(ll) for almost ally E ll'-1 •
Let k E BVJ and let l (k2 , ••• , kq)· By applying integration by parts and the Cauchy-Schwarz inequality we deduce
j lxk(otf) (xWdx = j y21(j x~"'(8tf) (xtoy) (8tf) (x~>y)dxt)dy R• Ro-1 R
= j y21(j f(x~ov} (2ktxi"''-1(8tf) (xt,y)+xi"'(oif) (xtoy))dxt)dy R•-1 li
Hence otf satisfies assertion (iii). Similarly, 83/ satisfies assertion (iii), j = 2, ... , q. Now an inductive procedure yields that f satisfies assertion (ii). Finally, suppose f satisfies assertion (vii). For yE Bq-l and k, lE BV0 we have
sup lx~ /(xt,Y)I $ sup lxU(x)l < oo, x1ER xeli9
sup 1(8~/) (xhy)l $ sup 1(8~/) (x)l < oo. :r,ER zEii•
That is, according to Theorem 1.2:
33
The function x1 ~----+ l(x11 y) E S(JR) for ally E /R'I.:..1•
Furthermore, we have I E £1(/R'I) because
/ ll{x)l dx :5 sup 1(1 + lxl2) 9 l(x)l j (1 + lxl2
)-9 dx . ~ ~~ ~
By use of the latter two results we estimate for k,l E IN0 ,
j <J 1(~1) (xt.Y)I2 dxt)dy = (-1)1 j <J (lf;11) (xt,y) l(xhy)dxt)dy ~-1 R Rq-1 R
:5 sup l(lf;11) (x)l j ll(x)l dx. :tER• R•
Hence xU E L2(JR9) and 8if E L2(JR:I). Similarly xjl E L2(JR:I) and ajl E L2(/R9),
j = 2, ... ,q. Therefore, I satisfies assertion (v) and the proof is complete. 0
As a straightforward consequence of Theorem 1.23 we obtain the following functional analytic characterization of S ( JR:I):
Theorem 1.24 q
S(JR'~) = D00 (Q) n D00(P) = D""(IQI) n D00(IPI) = n [D00(Qi) n D""(Pi)]. j=l
The equalities hold as topological vector spaces.
The operators IF,IFi, Pi and Q1 map S(/R9 ) continuously into itself. By Theorem 1.24 and the relations (1.48), (1.49) we have
IF(S(IR9 )) = S(JR:I) , IFj{S(IR'~)) = S(JR:I) , j 1, ... , q. (1.52)
The space S(JR:I) can be characterized by means of the (partial) Fourier transformation(s) (the proof runs similar to the proof of Theorem 1.5):
Theorem 1.25 Let I E £ 2( JR'~) and let p E { 2, oo}. The following assertions are equivalent:
(ii) x" I E Lp(JR:I) and x11F I E L"( IR9 ) , k, lE INri ;
(iii) lxi"J E Lp(JR:I) and lxi11Ff E L"(JR:I), k,l E /No;
(iv) xjf E Lp(JR:I) and x~IF;/ E Lp(JR:I), j = 1, ... ,([, k,l E /N0 •
34
The space S(JR.'l) can be characterized in terms of the Hermite expansion coefficients of its elements. The Hermite functions on R'~,
(1.53)
constitute an orthonormal basis in L2 (R'~) and they are eigenfunctions of the operator IPI2 + IQI\ cf. (1.11),
(IPI 2 + IQI2) Ill,. = (2lnl + q) Ill,. , n E IN3 , (1nl = n1 + ... + nq) . (1.54)
By applying the same techniques as in the proof of Theorem 1.6 we arrive at the announced characterization of the space S(R'~):
Theorem 1.26 q q
(i) S(R'~) = n ID""(QJ) n D""(P;)] = n D""(PJ + Qj) = D""(IPI2 + IQI2) .:i=l .:i=l
= {/ E L2(JR.'l) (nj(f, W,.)L2(~)) E loo , k E /No , j = 1, .•• , q} ,
= {/ E Lz(JR.'l) (lnlk (/, 111n)L2(~)) E loo , k E .OVo} .
The equalities hold as topological vector spaces.
(ii) For f E S(JR.'l) we have
f(x) = L (f, Wn)L2(Rq) 11/,.(x), X E R'l, nEN3
where the series converges in S(R'~). The series converges absolutely with respect to the /;oo(R'~)-norm and the series converges uniformly on R'~.
1.5 Spherical harmonics
In this section we present an introduction to spherical harmonics. The results will be used in subsequent sections where we use expansions in spherical harmonics in S(R'). We equip the vector space R'~ with the Euclidean inner product x · y x 1y1 ~ , •• + XqYq,
x, yE /R'~. Let et. ... , eq be the standard basis in JR.'l, that is, (ei)k = li;,k , j, k = 1, ... , q. The unit sphere in IR'~ is denoted by sq-t and its elements by Greek symbols{, q,w, .. . . Each x E IR'~\{0} can be uniquely written as x = rw with r the length of x, r = lxl, and w its direction, w = x/r E sq-l. Usually r is called the radical part of X a.nd w the spherical part of x. There exists a. measure a'~- 1 on the unit sphere S'~- 1 such tha.t
j f(x)dx = j j f(rw)rq-t drda9- 1(w), f E L1(R), q "2: 2. (1.55) Rq R+ sq-l
We gather some well-known properties of the measure aq-t.
Theorem 1.27 Let q "2: 2. (i) Let g E L1 (Sq-t ). Then the following recursive integration formula holds for q "2: 3:
.. _ I g(w)duq-I(w) =I I g(cos cp, {sin cp) sinq-2 cp dcpdu9-2
({),
so-l o so-a
with
2.-1 g(w)du1(w) =I g(cos cp, sin cp)dcp. Sl 0
(ii) The measure uq-l is a finite measure with
0 9 = uq-1(89- 1) = 2r912/f(q/2).
(iii) Let g E L1(Sq-l) and let A E JR:lXq be an orthogonal matrix. Then
I g(Aw)duq-1(w) = I g(w)du9-1(w). so-l so-l
35
(iv) Let h be a continuous function on [-1, 1], let { E sq-l and let A E JR911. 9 be an orthogonal matrix such that Ae1 = {. Then
.. I h(w ·{)du9-1(w) = 0 9_ 1 I h(cos cp) sinq-2 r.p dcp.
so-l o
Proof. (i) Let f(x) = exp(-lxl) g(x/lxl), x E IR9\{0}. Then f E L1(/Rq) and by (1.55)
I f(x)dx=r(q) I g(w)du9-1(w).
R• so-l
By using (1.55), with q replaced by q -1 and f by y t-t f(x 11 y), and by introducing polar coordinates in the (x11 r )-plane thereafter, we deduce
I f(x)dx =I (I f(x 11 y)dy)dx1
RO B R•- 1
=I <I I exp(-Jxi+r2)g((xt,rw)/Jxi+,·2)7·9-2 drduq-2(w))dx1
R B+ so-2
.. = r(q) I I g(cos cp, w sin r.p) sinq-2 <.p dcpduq-2(w).
0 S0-2
By identifying these two results, we obtain the stated recursive integration formula. For g E L1(S1
) we have, by (1.55) with q = 2 and by using polar c.oordinates in /R2,
2lf
I g(w)du1(w) =I exp(-lxl) g(xfixl)dx =I g(cos <.p, sin r.p)dr.p. Sl R2 o
(ii) From assertion (i), with g(w) = 1, we obtain
36
.. n, = !lq-1 J sinHllf' dc.p = nq-1 .,fi r((q- 1)/2) I f(q/2) with nz = 211".
0
Now an induction procedure yields the wanted result. (iii)'The assertion is a consequence of the property of the Lebesgue measure
j f(Ax)dx = j f(x)dx, f E L1(1Rq), A an orthogonal matrix in JRq><q. R9 R9
(iv) Observe that Aw · { Aw · Ae1 w ·et. wE Sq-:-1• So by means of assertion (iii) with g(w) = h(w. e), assertion (i) with g(w) = h(w. et), and assertion (ii), we deduce
.. J h(w. e) daq-l(w) j h(w · et)da9- 1(w) = !lq-1 j h(cos If') sin9- 2 If' d!.p. sq-t 5,-1 o
A polynomial p on /Rq is called
harmonic if f:.p = 0, and
0
homogeneous of degree m (or m-homogeneous) if p(.Ax) = Amp(x) , A> 0, x E JR:l.
From Miiller [39,, Definition 1] we quote the definition of spherical harmonics.
Definition 1.28 A spherical harmonic of degree m in q dimensions is the restriction to sq-l of an m-homogeneous harmonic polynomial on JR:l.
By )'! we denote the space of all spherical harmonics of degree m in q dimensions. A spherical harmonic Ym E ~ has the properties
Ym(-w) = (-l)m Y,.(w), wE sq-t, Ym has a unique harmonic extension Ym defined by Ym(rw)=r"'Y .... (w), r>O, wesq-1 .'
We equip~ with the L2(Sq-1)-inner product. In Faraut [21, pp. 8,9] the following wellknown results have been proved.
Theorem 1.29 (i) The space Y~ is a finite-dimensional Hilbert space of dimension N(q, m) given by
N(q,m) = (2m+q-2) r(m+q 2)/(f(q-1) f(m+ 1)).
(ii) Fork f. I, the spaces yz and Y't are orthogonal subspaces of L2(S"-1) and the linear span of the spherical harmonics is dense in L2(Sq-t ):
00
Lz(sq-1) = Ef1 Y!, .
m=O
In the remainder of this section it is understood that q ? 3. A spherical harmonic Ym E Y~ is called zonal with pole { E sq-1 if for all orthogonal matrices A E JRqXq with the property Ae = e we have
37
It turns out that a zonal spherical harmonic can be expressed in terms of the Gegenbauer polynomial 0!{2- 1 defined by, cf. Erdelyi et al. [18, 11.1.2 (25)],
oq/2-l(x) = ( -2)"' r(m + q/2- 1) r(m + q- 2) (1- x2t(q-3)/2
m m! r(q/2-1) r(2m+q-2)
(dfdxr [(1- x2)"'+(q-a)f2] , lxl :5 1 .
Indeed, in [18, 11.2 Lemma 1] the following well-known result has been proved:
Lemma 1.30 Let Y,. E y:, be a zonal spherical harmonic with pole { E S'-1 • Then there exists ~ E C such that
Y,.(w) = ~0!{2- 1 (w · {) , wE 89-1
•
Of special interest is the Funk-Hecke theorem, see [18, p. 247]:
Theorem 1.31 (Funk-Hecke) Let f be a continuous function on [ -1, 1J and let Ym E Y:,. Then
j f(w · {) Ym(w) du9-1(w) = ~ ... Y,.(e) , { E SH ,
s•-' with
1
~m (flq-t/0!{2-1(1)) J j(t)O!f2-1(t) (I- t2)(q-3)/2dt.
-1
We mention two applications of the Funk-Hecke theorem:
Theorem 1.32 Let Y,. E Jl!, let r > 0 and let { E sq-t. Then
(i) j O!f2-
1(w·{) Ym(w) do-9-1(w) = (f!90!f2
-1(1)/ N(q,m)) Y.n(e),
Sq-1
(ii) j exp(-ir(w·{)) Y,.(w) du9- 1(w) = (-i)m (21r)qf2 r·-q/2+l J,+9t2- 1(r) Y,.({).
s•-'
Proof. (i) Take f qp-t in Theorem 1.31. Then we obtain the stated integral formula by means of Theorem 1.27 (ii), Theorem 1.29 (i) and the formulas [18, 11.1.2(26) and (28)]
1
J [C'~/2-t( )]2 (I _ 2)(q-3)/2 d = 24
-9
11" f(m + q 2) m t t t m!(2m+q-2)[f(q/2-l)j2'
-I
(1.57)
0!{2- 1(1) f(m+q 2)/(m!r(q-2)). (1.58)
38
(ii) Take f(t) = exp(-irt) in Theorem 1.31. Then we obtain the stated integral formula by means of Theorem 1.27 (ii), relation (1.58) and the integral formula [38, p. 221]
11:
j exp(iz cos 9) C!{2- 1(cos 9) sin'~-2 1J dO 0
im ...(if(q/2- 1/2) r(m + q- 2) (2)q/l-l r(q- 2) r(m + 1) ; Jm+q/2-l(z) . (1.59)
D
Since~ is a. finite-dimensional Hilbert space, there exists a reproducing kernel K! : sq-l X
S 11- 1 -+ c of~. uniquely defined by the conditions:
1. For { e 8 11-1 the function w ~--+ K!(w,{) belongs toY!;
2. for { E sq-1 and Ym E Y! we have Ym({) j Ym(w) J<~,(w,{) dO'H(w); s•-•
see Aronszajn [l, pp. 343, 346]. From Lemma 1.30 and Theorem 1.32 (i) we deduce the explicit expression for the reproducing kernel K! ofY!:
As a consequence we obtain the L00-estimate for Ym E ~.
IY,.(e)l2 :::; IIYmii~(S•-•) K!({,{) = (N(q,m)f!lq) IIYmii~(S•-•), e E sq-l. (1.61)
In the paper [2] by Van Berkel and Van Eijndhoven, various L00 - and L2-estimates for (partial derivatives of) spherical harmonics have been established. For later use, we mention one of them [2, Theorem 8(ii)]: For I E IN6 and Ym E ~.
I((Jl ~ ) ( )12 2111 N(q, m- Ill) m! r(m + q/2) 11 112 sq 1
Y,.,. e :::; !lq(m- Ill)! r(m + q/2 -Ill) Ym ~(S•-•) ' { E - . (1.62)
From Theorem 1.32 (ii) we obtain the Hecke-Bochner theorem, cf. [21, Theoreme IUO]:
Theorem 1.33 (Hecke-Bochner) Let g: [O,oo) -t C such that g E L1(JR+;rm+q-1 dr), let Ym E ~ and define f : IR11 -t C by
f(rw) = g(r) Ym(rw)' r ~ 0, wE sH . Then the Fourier tmnsform IF f is given by
(IFJ) (rw) = (-i)"' (/Hm+q/2-19) (r) Ym(rw)' r 2 0' wE sq-l.
Thus we see that the Fourier transform of a. radially symmetric function f(rw) g(r), with g E L1(JR+; rq-l dr), is also radially symmetric and can he expressed as a Hankel transform,
(IF f) (rw) = (/H11/2-t9) (r), r 2 0, wE 8 9-1
• ( 1.63)
39
The latter formula. can also be derived without using the F\mk·Hecke theorem. Indeed, it readily follows by use of (1.55), Theorem 1.27 (ii) and (iv), and (1.59) with m= 0.
Next we shall introduce an orthonormal basis in L2(/R'I) of eigenfunctions of the oper· ator IPI2 + IQI2
, which differs from the Hermite basis lli,., nE IVJ, cf. (1.54). For that purpose we establish some auxiliary results. We define the differential operator x · V and the momentum operator M by
q
x ·V= L x/}; and M= L (x;f]J.- x~c8;)2 • (1.64) j=l l~i<k~q
Lemmal.S4,
(ii) MYm -m( m+ q- 2) Ym 1 Ym E }':,;
(iii) Let g E 0 2(/R), let hE C2(/R9) and define f E C2(1R9\{0}) by f(x) = g(lxl) h(x), x E JR'I. Then for x E JR'I\ {0},
(M f) (x) = g(lxl) (Mh) (x).
Proof. (i) The identity follows by a straightforward calculation. (ii) Let _Ym E Y~. Since the pol~nomial Ym is homogeneous of degree m, we have (x ·V) Ym = mYm. Now, since Ym is harmonic, it follows from assertion (i) that
(iii) For j E {1, ... , q} we have
(8if) (x) = (xi/lxl) g'(lxl) h(x) + g(lxl) (8ih) (x), x E IR9 \{0}.
Hence for 1 ~ j < k ~ q,
The wanted identity is now obvious. 0
Let {Ym;.; : j = 1, ... , N(q, m)} be a.n orthonormal basis in Y,~., fixed throughout the thesis.
Theorem 1.35 Define the functions u,.,m,j : JR'I -+ C by
Un,m,j{rw) = L;:'+q/Z-l(r) Ym,j(rw), n,m E IVo, j = I, ... ,N(q,m),
,. ~ o , w e sq-J .
Then the functions un,m..i constitute an orthonormal basis in L2 ( /Rq) and
40
(i) Mu,.,m,j = -m(m+q-2)u,.,m,j, n,m E DVo, j = 1, ... ,N(q,m),
(ii) (IPI2 + IQI2 )un,m,j = (4n + 2m + q)u,.,m,j, n,m E DVo, j = 1, ... ,N(q,m),
(iii), llun,m,;IIL.(no) :5 ( 2n + ~-+1 q-
1 ) , n, mE DVo, j = 1, ... , N(q, m).
Proof. The spherical harmonics Ym,j , m E DV0 , j = 1, ... , N(q, m), constitute an orthonormal basis in L2(Sq-1 ), by Theorem 1.29 (ii). Likewise, for fixed m E DV0 , the functions r 1-+ rm JL:;'+q/2- 1(r) , n E DV0 , constitute an orthonormal basis in Xq-h by (1.12). By means of Parseval's identity and Fubini's theorem we have for f E L2(/Rq),
11/lli.(no) = j ( j lf(rwW duq- 1(w)) rq-1 dr n+ so-l
oo N(q,m) oo = L L L I j ( j f(rw) Ym,;(w) duq- 1(w)) 1·'-" JL;:'+q/271 (r) rq-1 drl 2
m=Q J=1 n=O R+ so-l
oo N(q,m) = L L I j j f(rw) JL;:'+q/2- 1(r) Ym,;(rw) rq-1 dr duq-1(wW.
m,n=O J=1 R+ so-l
Hence the functions Un,m,i• n, mE DV0 , j = 1, ... , N(q, m) constitute an orthonormal basis in L2(1Rq). (i) This identity is a consequence of Lemma 1.34 (ii) and (iii). (ii) Observe that x ·V = rofor. By means of Lemma 1.34 (i), identity (i) and Corollary 1.13 (ii), we deduce for n, mE DV0 , j = 1, ... , N(q, m), r > 0, wE Sq-1 ,
((IPI 2 + IQI2)un,m,j) (rw)
= ( -82 for 2- (q- 1) r-1 of or- r-2 M+ r2) JL;:'+q/2-
1(r) Ym,i(rw)
= Ym,;(rw) r-m( -~ /dr2- (q- 1) r-1 d/dr +m( m+ q- 2) r-2 + r2)
= Ym,;(rw) ( -d2 /dr2- (2m + q- 1) r-1 dfdr+ 1·
2) JL;:'+q/2
-1(r)
= (4n+2m+q)JL;:'+q/2-
1(r)YmJ(rw) = (4n+2m+q)un,m.;(rw).
(iii) Since both {wk: k E llVJ} and {un,m,; : n,m E JJV0 , j = I, ... ,N(q,m)} are orthonormal bases in L2 (1Rq) of eigenfunctions of the operator IPI2 + IQI\ cf. {1.54) and identity (ii), we have for n, mE JJV0 , j = 1, ... , N(q, m),
Un,m,j = kEN~:Ikl=2n+m
41
and therefore, by the Cauchy-Schwarz inequality and the estimate (1.10),
lun,m,j(x)j2::; llun,m,iii~(R•)
keN3:1kl=2n+m
0
Finally, we introduce the Laplace-Beltrami operator tl.Ls in L2 (S9- 1 ), defined on the domain
(1.67)
by
oo N(q,m)
tl.LB g = - E E m( m+ q- 2) (g, YmJ)i4(S•-•) Ym,j . (1.68) m=O i=l
The operator ALs is self-adjoint and due to Lemma 1.34 (ii) we have for m e /N0 , j = l, ... ,N(q,m),
(ALsYmJ) (w) =-m( m+ q- 2) Y.,.,j(w) = (MY.,.J) (w) , we SH . (1.69)
1.6 The space S(Jiri) and expansions in spherical harmonics
We continue our discussion of the space S( JR'I). First we characterize the radially symmetric functions in S(JR'I). Thereafter, we consider products of radially symmetric functions and homogeneous harmonic polynomials in S(Dl9 ). As an important result we show that a function f e S(Dl9) admits the following type of expansion,
oo N(q,m)
f(rw) L E fmJ(r) Ym,;(rw) ' r ~ 0 , we sq- 1 ,
m=O i=l
where the functions fmJ , me /No, j = 1, ... , N(q, m), belong to Seven(Dl) and where {Ym,i : j = 1, ... , N(q, m)} is the orthonormal basis in Y,~,, introduced in Section 1.5; see Theorem 1.39. In the second part of this section, we introduce the space S(Zq) and we identify S(Dl9 )
with a subspace of S(Z9), see Theorem 1.43. By means of this characterization we can solve the following factorization problem in Theorem 1.44: For which functions g and h does the function (rw) ~---+ g(r) h(w) belong to S(Dl9 )? The space S(Z9 ) reappears in the next section, where we study the Radon transformation on S(Dl9).
Theorem 1.36 Let g : [O,oo) --+ C such that g E Xq- 1 and define the radially symmetric function f : JR'I --+ C by
f(rw) = g(r), r 2:: 0, wE 8 9-1
•
Then f e S(Dl9 ) if and only if g e Seven(Dl).
42
Proof. According to Theorem 1.25, the function J belongs to S(JR'I) if and only if
lxlk J E Loo(JR'I) and lxi11FJ E Loo(JR'I), k,l E lNo.
That is, by (1.63), J E S(JR'I) if and only if
r~<g E L00(W) and r 1 Dlqf2-t9 E Loo(ll+), k,l E /No.
Now the wanted equivalence is obtained from the characterization of Seven(ll) in Theorem 1.12. D
Theorem 1.37 Let g: [O,oo)-+ C such that g E X2m+q-lt let Ym E.~ and define J: ll9 -+ c by
J(rw) = g(r) Ym(rw)' r 2 0' wE sq-l. Then J E S(llq) if and only if g E Seven(ll).
Proof. Suppose g E S.ven{ll). Define the radially symmetric function iJ : JR'I -+ C by
iJ(rw) = g(r) , r 2 0 , wE 59-1
•
Then J(x) g(x) Ym(x), x E JR'I. Hence J E S(JR'I), because Ym is a polynomial and g E S(ll9 ) by Theorem 1.36. Conversely, suppose J E S(JR'I). Since Ym is bounded on S9 - 1 , it follows from Theorem 1.25 and Theorem 1.33 (Hecke-Bochner), as in the proof of Theorem 1.36, that
r"+m9 E L00(W) and rl+m IHm+q/2-19 E L00(1l+) , k,l E lNo .
Then we estimate for p > 0, by means of (1.16) and (1.21),
00
I(1Hm+q/2-t9) (p)l I j (rp)-(m+q/2- 1) Jm+q/2-1(rp) g(r)r2m+q-t drl 0
00
::; sup le-(m+q/2-1) Jm+q/2-1(e)l j jg(r)l (1 + r 2 ) r 2m+q-t(l + r2)-1 dr e>o 0
That is, IHm+q/2-19 E L00(1l+). Observe that JH!,+q/2_1g = g, by (1.20). So by replacing g by 1Hm+q/2-t9 in the above estimation, we also have 9 E L00(1l+). Thus it follows that
rkg E L00(1l+) and r 11Hm+q/2-19 E Loo(W) , k, lE !No.
Then the characterization of Seven(IR) in Theorem 1.12 yields g E Seven(IR). 0
The space S(ll'~) can he characterized in terms of the expansion coefficients of its elements, with respect to the basis {un,m,j, n,m E /No, j = l, ... ,N(q,m)}, introduced in Theorem 1.35.
43
Theorem 1.38
(ii) For f E S(D/:1) we have
oo N(q,m)
f(x) = L: L: (!, Un,m,j)~(.Rt) Un,m,;(x) , X E JR'I , n,m=O i=l
where the series converges in S(D/:1). The series converges absolutely with respect to the L00(IR'~)-norm and the series converges uniformly on D/:1.
Proof. (i) This assertion follows from the characterization of S(IR'~) in Theorem 1.26, viz.
and the fact that the functions Un,mJ constitute an orthonormal basis in L2( IR'~) such that
(IPI 2 + IQI2) Un,mJ (4n + 2m + q) Un,m,j ' n, me /No. j = 1, ... , N(q, m)'
cf. Theorem 1.35 (ii). (ii) By applying the same techniques as in the proof of Theorem 1.6 (ii), it follows that the mentioned series converges to f(x) in D00(IPI2 + IQI2 ) = S(JR9 ).
The remaining part of assertion (ii) readi]y follows from the estimate in Theorem 1.35 (iii): Forn,me.JN0 , j 1, ... ,N(q,m),
llu ·ll2 < ( 2n +m+ q -1 ) < (2n +m+ q -l)q-1
n,m,J L..,(ID)- q-1 - (q-1)! · 0
Theorem 1.39 Let f e S(JR9) and define the functions f,.J : [0, oo) --+ C, m E .JN0 , J = l, ... ,N(q,m) by
00
fmJ(r) = L: (!, Un,m,;)L2(.R') JL;:+q/2-l(r) , r :2: 0 . (I. 70) n=O
Then (i) fm.i E Seven(IR), mE /No, j = l, ... ,N(q,m), and tht: series in (1.70} converges in Seven(IR). The series converges absolutely with respect to the L00(JR+)- norm and the series converges uniformly on JR+;
oo N(q,m) (ii) f(rw) = L: L: fmJ(r) Y,.,j(rw) , r :2: o , we s'~- 1
,
m=O i=t
where the series converges in S(JR'~). The series c.onvergcs absolutely with respect to the L00 (1R9 )-norm and the series converges uniformly on /R9 ;
(iii) r"' fmJ(r) = j f(rw) YmJ(w) da'~- 1 (w) , r > 0 , m E /No , sq-t
j = l, ... ,N(q,m}.
44
Proof. (i) The assertion readily follows from Theorem 1.38 (i) and the characterization of Seven(ll) in Theorem 1.8. (ii) The assertion follows from (i) and Theorem 1.38 (ii). (iii) The assertion follows from (ii) and the orthogonality relations for the functions Ym,i· D
The remainder of this section deals with some preliminaries to the next section, where we study the Radon transformation on S(IR'). We introduce some notations. Z9 denotes the unit cylinder in Jl9+1, defined by
Z9 = 11. x sq-l •
L2(Z9) denotes the Hilbert space of square integrable functions on Z9 with inner product
(f,g)L2(Zq) = j j f(p,w) g(p,w) dpd0'9-1(w) .
R sq-1
For f E L2{1l) and g E L2(S9- 1) we define f ® g E L2(Z9 ) by
(!®g) (p,w) = f(p) g(w), (p,w) E Z9 •
For self-a.djoint (unitary) operators T1 and T2 on L2( ll) and L2( S11- 1 ), respectively, we introduce the tellSOr product. T1 ®T2 on L2( Z9 ) as the unique self-a.djoint (unitary) extension of the operator T, defined on the domain span{!® g: f E D(Tt), g E D(T2 )} by
n n
T(L'.: ci(fi ® Yi)) = L: ci(TtfJ) ® (TzgJ) , j=1 j=l
n E IN , CJ E C , fJ E D(Tt) , gi E D(Tz) ,
cf. Weidmann [50, Section 8.5]. Since the Hermite functions Tjl,., n E /N0 , constitute an orthonormal basis in L2(JR.) and since the spherical harmonics Ym.J , m E /N0 , j = 1, ... , N(q, m), constitute an orthonormal basis in L2(S9-
1), the functions TjJ,. ® Ym.J , n, mE JN0 , j = 1, ... , N(q, m),
constitute an orthonormal basis in Lz(Z9 ).
For the remainder of this section, the following operators are of special interest:
IF® I, Q ®I, P ®I, (P2 + Q2) ®I and I® D.Ls .
Let f E S( Jl9+l) and let g be the restriction of f to Z9• Then we observe that for (p,w) E Z9 ,
((IF® /)g) (p,w) = ~ j g(t,w)e-i"1dt, v21r R
((Q ®I) g) (p, w) = pg(p,w) ,
((P ®/)g) (p,w) = i(8f8p) g(p,w),
((P2 + Q2) ®I) g) (p,w) = ( -az;apz + p2) g(p,w) '
((/ ® D.Ls)g) (p,w) = (D.LBg(p,·)) (w).
45
Definition 1.40
Let us give a motivation for this definition. Functions g E S(Z9 ) have the following properties:
I. For fixed wE sq-t, the function p ~-+ g(p,w) belongs to S(IR} (because of Theorem 1.3 and g E /JOO(Q 0 /) n D00 (P 0 /));
2. for fixed p E JR., the function w ~-+ g(p, w) belongs to C00 ( Sq-l)
(because g E D00(10 .6-LB )).
In fact, S( Z9 ) consists of the restrictions offunctions in S( JR:l+l) to Z9 •
Theorem 1.41
(i) S(Z9 ) = n D(Qkpl 0 .6.'£8 ) = D 00((P2 + Q2) ®I) n D00(/0 .6-Ls) k,l,mENo
= {g E L2(Z9): (nkm1(g, tPn ® Ym,j)L,(Z,))n,m,j E loo , k, lE lNo} .
The equalities hold as topological vector spaces.
(ii) ForgE S(Z9 ) we have
oo N(q,m)
g(p,w) = 2: 2: (g, tPn 0 Ym,j)L,(Z,) tPn(P) Ym,j(w), (p,w) E Z9 ,
n,m=O i=l
where the series converges in S(Z9 ). The series converges absolutely with respect to the L00(Z9 )-norm and the series converges uniformly on Z9 •
Proof. (i) By using the characterizations of S(IR) in Theorem 1.2 and 1.6 (i) we derive
k,l,mENo k,l,mENo
= n (D(Q" 0 /) n D(P1 ® /) n D(J ® .6.~'8 )) k,l,mENo
= D00(Q ® /) n D00 (P ®I) n D00(/ ® ~LB) = S(Zq)
= D00((P2 + Q2) ® /) n D00(i 0.6-La)
= {g E L2(Zq) : (nk(g, tPn ® Ym,j)L,(Z,))n,m,j E loo , k E /No ,
(m21 (g, tPn ® Ym.j)L,(Z9 ))n,m,j E l,x:, , lE b'Vo}
= {g E L2(Z9 ): (n"'m1(g, tPn 0 Ym,;)L2(Z0))n,m,j El"" , k, lE lNo} .
46
(ii) By applying the same techniques as in the proof of Theorem 1.6 (ii), it follows that the mentioned series converges to g(p,w) in D00((P2 + Q2 ) 0 /) n D00
(/ 0 ~LB) = S(Zq)· By the inequalities (1.10) and (1.61) and by Theorem 1.29 (i) we have
n~ .. IIL«>(R) :5 1 I n E /No I
IIYmJIIL(sq-1) :5 N(q,m)/ftq
= (2m+q-2) f(m+q-2)/(ftqf(q-1) r(m+ 1))
:$2(m+q/2-1)9-2 /(ft9 f(q-1)), mEIN0 , j=1, ... ,N(q,m).
The remaining part of assertion (ii) readily follows from these estimates. 0
A function g is even on Z9 if g(p, w) = g( -p, -w), (p,w) E Zq. The space Seven(Z9 ) is the subspace of even functions in S(Z9 ).
Let f be a function on /R9 • Then V f is the function on Zq defined by
(V f) (p,w) = f(pw) , (p,w) E Z9 • (1.71)
Obviously, V f is even on Z9 •
In the next theorem we show that the space S(JR9) can be regarded as' a subspa.ce of Seven(Z9 ). Moreover, we establish the connection between the momentum operator M and the Laplace-Beltrami operator I 0 ~LB, cf. (1.69).
Theorem 1.42 Let f be a function on JR9. (i) If f E S(/Rq), then V f E Seven{Zq} and for lE /No,
(M1J) (rw) ((I® ~~s) V!) (r,w) I r > 0 I wE sq-l.
(ii) If V f E Seven(Z9 ), then f E C00 (1Rq\{O}) and forl E /No,
(M1J) (rw) ((10 ~~8 ) V!) (r,w), r > 0, wE ,(j'/-1 •
Proof. (i) Suppose f E S(/Rq). Then for k E IN we have
j j ll f(pw)l2 dpduq-1(w) R Sq-1
j j (1 + P2tt ((1 + p2)p2klf(pw)l2)dpduq-l(w) R sq-1
That is, V f E D""(Q ®I). Since ((P ®I) V f) (p,w) = w1(otf) (pw) + ... + wq(Oqf) (pw) , (p,w) E Z9 , there exist constants ak,l E /R, I E IN, k E INJ with lkl = 1, such that for I E IN,
((P1 ®I) V f) (p,w) = L a~c,t wk(Qk f) (pw) , (11,w) E Z9 • keN:.Ikl=l
( 1. 72)
47
For m, n E INJ we have
j j lw"'(8m f) {p:.l)wn (8" f) (pw)l dpdu9-1(w) R St-1
From the latter two results we conclude that V f E D00(P ® /). Thus we have shown that V f E D00(Q®l)nD00(P®l). This implies (by Theorem 1.3) that for fixed w E sq-l I the function p 1-+ (V f) (p, w) belongs to S(/R.). By using Theorem 1.6 (ii}, we then deduce for r > 0, mE /N0 , j = 1, ... , N(q, m),
j /(re) YmJW dc79- 1({) = j <f </ f(t{) t/Jn(t) dt) t/Jn(1·)) Y,.,;(e) dc79- 1(e) St-1 St-1 n=O R
= f: j j f(te) 1/J,.(t) Ym,j({) dt dc79-1({) tPn(r) = f: (V/, tPn ® YmJ)L2(Z,) tPn(r).
n=O R St-1 n=O
The momentum operator M maps S( /8:1) continuously into itself. Let r > 0, w E S9- 1 and l E /N0 • By means of Theorem 1.39 (ii), (iii), Lemma 1.34 (ii) and the previous result it follows that (M1f) (rw) equals
oo N(q,m)
( -1)1 L ?: (m( m+ q- 2)}1 j /(r{) Ym,;({) de1q-1({) Ym,;(w)
m=O .1=1 $<1-1
oo N(q,m)
= (-1)1 L L (m(m+q-2))1 (Vf,t/Jn®Y,.,;)L:~(Zq) .,Pn(r)Ym,;(w), n,m=O j=l
where the series converges absolutely with respect to the L00(/R'I)-norm. Hence V f E D00 (1 ® L\Ls) and
(M1f) (rw) =((I® L\~s) V f) (r,w), r > 0, wE 8 9 -1
.
This completes the proof that V f E Seven(Z9).
(ii) Suppose V f E Seven(Z9). According to Theorem 1.41 (ii) we have
eo N(q,m)
/(x) = L L (V/,1/Jn ® YmJ)L:~(Z,) lxl-m tPn(lxl) Y,,.,;(x), X E Rl\{0}. n,m=O i=l
By means of the estimates (1.10) and (1.62), we have for k, n, p E 1N0 , l E lNJ, m ;:: Ill, j = l, ... ,N(q,m),
llrktfJi"llli..(R+) :5 2k+"(n + k + p)! / n! :5 2k+"(n + k + IJ)k+p ,
llai- ll 2 2111 N(q, m -Ill) m! r(m + q/2)
Ym,j L .. (s•-•) :5 !l9(m -Ill)! f(m + q/2 -Ill)
:5 21'1+1 (m -Ill+ q/2- 1)9-2 m111 (m + q/2 -1)111 / (Hq f(q- 1)).
48
Let e > 0 and let G~ = {x e JR'l : lxl > e}. Then it follows from the previous estimates and Theorem 1.41 (i) that for I e JN3,
oo N(q,m)
, L L II(V f, t/J,. ® Ym,j)L2(Z9 ) 01(lxl-m t/J,.(Ixl) Ym,;(x))IILoo(Gc) < 00 • (1.73} n,m=O j=l
Hence f e C""(Dl9\{0}) and, by Lemma 1.34, (M1!) (rw) equals
oo N(q,m) (-1)1 L L (m(m+q-2))1 (Vf,t/J,.®Ym,j}L2(Z.) 1/J,.(r) Ym,;(w)
n,m=O j=l
=((I® 6~8) Vf)(r,w), r > 0, we S'~- 1 , I e JlV0 . 0
Let f be a. function on Dl9 such that V f e Seven(Zq)· Then we have proved in Theorem 1.42 (ii) that f e C""(Dl9\{0}). From the proof it is not hard to see that x"f, k e JN3, is bounded on JRq, while it follows from (1.73) that 81f, l e JN3, is bounded on Ge = {x e Dl9 : lxl > e}, e > 0. Hence, we may conclude from Theorem 1.23 that f e S(JR'l) if and only if f is infinitely differentiable at 0. An additional condition on V f is needed to ensure that f is infinitely differentiable at 0.
Theorem 1.43 'Let f be a function on JR9. Then f e S(JR9 ) if and only if (i) V f e Seven(Z9 ) and . (ii) for le JlV0 , the function w ..... ((P1 ®/)V f) {O,w) is the restriction to sq-t of a homogeneous polynomial of degree 1 on JRq.
Proof. Suppose f e S(JR9 ). Then f satisfies the conditions (i) and (ii) by Theorem 1.42 (i) and the identity {1.72). Conversely, suppose V f satisfies the conditions (i) and (ii). We shall prove that
lxlk f e L2{lR9 ) and 6 1 f e L2(JR'l) , k, le llVo .
Since V f e S(Z9 ) c D 00(Q ®/),it follows that lxl"f e L2 (JR'~), k e JlV0 •
Let I e JlV0 • We show that 6 1 f e L2(JR'l) by a separate investigation of the two integrals
j j !(61f) (rw)!2 r'~- 1 drdO'q-l(w), 1 s•-1
} j 1(61!) (rwW r 9-1 drd0'9-
1(w). (1.74) o s•-1
From the proof of Theorem 1.42 (ii) it follows that lxl{q+t)/2 6 1f, 1 e JN3, is bounded on {x e JR9 : lxl > 1}. Therefore, the first integral of {1.74) is finite. Next we consider the second integral of (1.74). Inspired by Helgason's paper [27, p. 162], we introduce the functions em, me /No, defined by
00
em(t) L ( -it)k / k! , t E lR, mE llVo . k=m
These functions have the obvious properties
e:,.(t) = -ie,._1(t), t e B, m~ 1;
m-1
e,.(t) = e-it- L ( -it)k I k! , t e B, m e /No ; k=O
lr"' e,.(t)l S e + 1 , t E B , m E /No •
By (1.76) we have for r > 0, we S 9-1,
f(rw) =(V f) (r,w) = ~ j e-'"((JF* ®/)V f) (t, w) dt v211" .R
21-1 ( • )k 1 = E -kt~ ((Pk ®I) V f) (O,w) + RC I e2,(rt) ((F• ®I) V f) (t,w) dt .
k=O • v211' .R
49
(I. 75)
{1.76)
(1.77)
By condition (ii), the sum in the latter expression forms a polynomial p on JR'I of degree 21- 1, so that t:J.1p = 0. Hence, by means of Lemma 1.34 (i) (with x ·V = r8f8r) and Theorem 1.42 (ii), we find
(fi1f) (rw) = ~ j (82/8r2 + (q -l)r-18/lJr + r-2fiLB)
1
v211' .R
(e2,(rt) ((F ®I) V f) (t,w)) dt .
By ( 1. 75) and ( 1. 77), there exist coefficients bj,k,t E C, j = 0, ... , l, k = 0, ... , 21, such that for r > 0, w e S9-
1 ,
I 21-2j
l(ti1f)(rw)l=l~ E bj,k,t I t21(rt)-(zl-k)ezt-k(rt)((F•®6t8 )Vf)(t,w)dtl 1=0 k=O .R
I 2l-2j
:5 (e + 1) E E lb;,k,tl f I((Q21 r ® 6tB) V f) (t,w)! dt. j=O k=O .R
Since ( Q21 F ® 6t8 ) V f E S( Z9 } , j = 0, ... , l, it follows that the second integral of (1.74) is finite. Hence t:J.1f e L2(B9), lE /N0•
We conclude that f E S(JR.'I), by Theorem 1.23. D
Let e.,. be the orthogonal projection of L2(S9- 1 ) onto~. that is
N(q,m)
e,.h = E (h, Y ... ,;h,(sH) YmJ' he Lz(S9-1).
i=1
We recall the definition of the space S ... (ot+),
S ... (JR+) = {g E S(ot+) : gl1l(o+) = 0 , le FV0 } .
We come to the following factorization result in S(JR'I).
(1.78)
50
Theorem 1.44 Let g : [0, oo) -+ C such that g E X 9_ 1 , let h.: S9- 1 -+ C such that hE L2(S9- 1 ) and define f: JR9-+ C by
f(rw) = g(r) h(w) , r ~ 0, wE Sq-t .
(i) if h = E!:'=0 ~mh with ~Mh-:/: 0, then f E S(JR9) if and only if g E Q2M(Seven(1R)).
(ii) If h = E!:'=o e2m+Ih with ~M+th-:/: 0, then f E S(JR9) if and only if g E Q2M+l(Seven(IR)).
(iii) If h cannot be represented by a finite expansion as in {i) or {ii), then I E S(JR9) if and only if g E s_(JR+) and hE D00(8Ls).
Proof. By means of Corollary 1.16 and Theorem 1.43 we have f E S(JR9) if and only if g E S(JR.+), h E· D00(8Ls) and, for l E !No, the function w .--+ g!1)(0+) h(w) is the restriction to sq-l of a homogeneous polynomial of degree l on JR9. Now the theorem follows by using the characterization of Q"(S.,ven(.D7.)) in Theorem 1.6a (ii) and by observing that YmJ can be extended to a homogeneous polynomial of degree m+ 2k, k E !No, on JR9 •
1. 7 The Radon transformation on S(JRn ·
The Radon transform of a function on JR9 is defined as the set of its integrals over the hyperplanes in JR9. This transform has its origin in Radon's paper [44] where he determines a function on /R2 from its line integrals. Nowadays, the Radon. transformation has found spectacular applications in medical imaging. We shall study the Radon transformation from a theoretical point of view. As general references we recommend Ludwig [37], Helgason [28], Deans (9] and Natterer [40]. We introduce some notations. The notation H = H(p,w) with (p,w) E Z9 stands for the hyperplane in JR:I given by
H = H (p, w) = {X E m' : X • w = p} .
Let f be a function on m', integrable over each hyperplane in /R9 • For (p, w) E Z9 , the Radon transform ('Rf) (p,w) is the integral off over the hyperplane H(p,w),
('RI) (p,w) = 1 f(x) dp.(x), (1. 79) H(p,wj
where p. is the Euclidean measure on the hyperplane H(p,w). The Radon transform 'Rf is even on Z9 , because H(-p,-w) = H(p,w), (p,w) E Z9 • For w E sq-J' let A.., E JRqXq be an orthogonal matrix such that A..,e) = w. We then have
('RI) (p,w) = 1 /(A..,(p,y))dy, (p,w) E Z9 • (1.80) R•-•
51
Lemma 1.45 Let f E S(.fl.'l) and let k : JR -+ C such that h E Loo(IR). Then for wesq-t,
j h(p) (R-I) (p,w)dp= j h(x·w) f(x)dx. R R•
Proof. For w E sq-t we have by (1.80),
j h(p) (R-I) (p,w)dp = j h(p) ( j f(A.,(p,y))dy)dp. R R R•-1
Now we obtain the wanted identity by the substitution x = A..,(p, y ), p = x · w, dpdy = dx. 0
An important result is the so-called projection theorem, which provides the fundamental relationship between the Radon transformation and the Fourier transformation, cf. Ludwig [37, (1.3)].
Theorem 1.46 Let f E S(JRil). Then
R-I= (211")<9-t)/2 (F* ®/)V F I.
Proof. For ( t, w) E Z9 we deduce by means of Lemma 1.45 with h(p) = exp( -itp ),
((F®/)'R.f) (t,w) = ~ j ('RI) (p,w) exp(-itp)dp y211" R
~ j f(x) exp(-it(x·w))dx = (21r){q-l)/2(Ff) (tw) y21r R•
= (211")<9-l)/2 (VFJ) (t,w)
where the operator V is defined by (1.71). 0
The range of the Radon transformation when applied to S(/Rq) has been characterized by Helgason [27, Theorem 4.1] by means of differential forms. h1dependently, Ludwig derived the same characterization by means of expansions in spherical harmonics, see [37, Theorem 2.1]. However, as observed by Helgason [28, p. 72], a crucial point in Ludwig's proof is missing and seems difficult to be settled in Ludwig's context. The characterization of V(S(R')) as a subspace of L2(Z9), implicit in Theorem 1.43, together with Theorem 1.46 immediately lead to the following characterization of 'R.( S( /R9 ) ).
Theorem 1.47 Let 9 be a function on Z9 • Then 9 E 'R.(S(.fl.'l)) if and only if (i) g E Seven(Z9) and (ii) g obeys the so-called Hel9ason-Ludwig consistency conditions: For l E /N0 , the function
W H J tl 9(t,w)dt is the restriction to sq-l of a homogen€:0US polynomial of degree f 011
R
52
Proof. Since JF(S(/R9 )) = S(IR9 ), see (1.52), we deduce by means of Theorems 1.46 and 1.43,
'R(S(/R9 )) =(IF*® I) V(S(JR9))
=(IF*® I) ({hE Seven(Z9): for lE /No, the function w 1-+ ((P1 ®I) h) (O,w) is the restriction to sq-1 of a homogeneous polynomial of degree l on JR9}) '
= {g E Seven(Z9): for lE /No, the function w 1-+ ((P11F ®/)g) (O,w) is the restriction to S9- 1 of a homogeneous polynomial of degree l on /R9 } •
The following observation completes the proof,
((P11F® I) g) (O,w) = ((JFQ1 ®I) g) (O,w) = vk j t1 g(t,w)dt. R
0
Our next aim is to derive Radon inversion formulas. To that end we introduce some operators. Msgn denotes the operator of multiplication by sg11(x) on L2 (/R) defined by
(M.gnf) (x) = sgn(x) f(x), x E IR, f E L2(/R),
where
sgn(x) = r 1 -1
if X~ 0,
if X< 0.
1i denotes the Hilbert transformation on L2(/R) defined by
1i = -iiF* M.gn IF .
Lemma 1.48 (i) (i1i) 2 = I ,
(ii) P1i = 1-lP ,
(iii) for f E S(/R) we have
(1.81)
'' (1.82)
(1.83)
(1-lf) (x) = 2~ j p- 1(/(x- p)- f(x + p)) dp = ~ f (x- Pt 1 f(p)dp, x E IR, R R
where f denotes that the Cauchy principal value is to be taken.
Proof. (i), (ii) The equalities are evident. (iii) As a preliminary we present the well-known integral formula
00
~ j p-1 sin(~p)dp = sgn(~), ~ E IR\{0}. -oo
for f E S( Ul) we derive, by using Lebesgue's dominated convergence theorem and Fubini's theorem,
53
('HI) (x) = ~ j sgn(e) (Ff) (e) et.:(~ v211' .R
1 00 1 00
= -2
j p-1(/(x- p) -f(x + p)) dp = - lim j p-1(/(x- p) -f(x + p)) dp 11' 11' ~10
-oo e
$- 00
= .!.. lim ( j (x- s)-1 /(s) ds + j (x- s)-1 f(s) ds) 11' c!O
-oo z+c
= ~ f (x- p)-1 f(p) dp , x E IR . 0
.R
We regard 'R. as a densely defined operator from L2(/R9 ) into L2(Zq), with domaiu S(JR9). For the adjoint operator 'R.* we have the following result.
Lemma 1.49 Let g : Zq -+ C such that g E L2(Z9 ) and, for w E sq-l, the function pH g(p,w) E Loo(IR). Then g E D('R.*) and
('R.*g) (x) = j g(x ·w,w)du9-1(w), x E JR9.
sq-1
Proof. Let f E S(JR9). By Lemma 1.45 with h(p) = g(p,w), we have
j <J ('R.f) (p,w) g(p,w}dp) du9-1(w) = j f(x) ( j g(x · w,w) duq-t(w)) dx.
Sq-1 .R R• Sq-1
Now we come to the following Radon inversion formula, cf. [37, Theorem 1.1].
Theorem 1.50 Let f E S(/Rq). Then
(i) I= ~(211') 1 -q 'R.*((i'H)q-t ( -P)q-1 ®I) 'R./
{
'R.*{(-P)q-t ® /)'R./ for q odd, = H211')1-q
'R.*(i'H(-P)q- 1 ® I)'R.f for q even.
0
54
(ii) For q odd we have
l(x) = H2'11'i)1-q 1 ((fJjfJp)H RI) (x ·w,w) do-9- 1(w), · x E JR:l. Sq-1
For q even we have
l(x) = 2(2'11'it9 1 p-1(ofop)9-1
( 1 (RI) (x · w + p,w) du9-1(w))dp, x e JR:l.
R+ 89-1
Proof. (i) By means of Lemma 1.49 we deduce for x E JR:l,
l(x) = (2'11')-912 1 <1 r 9-1 (Ff) (rw) exp(ir(x·w))dr·)du9-
1(w) s•-1 R+
= l(2'11')-q/2 1 <1 sgn9-1(r) r'-1(V F f) (r,w) exp(ir(x · w)) dr) du9- 1(w)
s•-1 n
The present result can be rewritten as
I= l(2'11')-,(q-l)/2 R.*((JF" M:;.l F) (F*QH F)®/) (F* ®I) V F I.'
Since P = -F*QF and 1t = -iF*M.gniF, cf. (1.5) and (1.83), we obtain th~ wanted equalities from Theorem 1.46 and Lemma 1.48 (i).
(ii) For q odd, the integral expression follows from assertion (i) and Lemma 1.49. For q even, the function g := ((-P)9- 1 ® /)'RI is odd on Z9• That is, g( -p, -w) = -g(p,w), (p,w) E Z9 • By means of Lemmas 1.48 and 1.49 we deduce for x E /R9 ,
(R.*(i'Jt(-P)q-t ® l)R.f) (x)
= 2~ I <1 p-1(g(x·w p,w)-g(x·w+p,w))dp)duH(w) sq-1 n
= ~i I p-1( 1 g(x · w + p,w)du9-
1(w))dp n s•-1
= 2(~i)'~ I p-1(ojfJp)'~-1 ( 1 (RI) (x ·w+p,w) du9 - 1(w))dp. n+ sq-1
The wanted integral expression then follows from assertion (i) .. 0
Observe that Theorem 1.50 (ii) with q = 2 equals Radon's original inversion formula, see [44, A. Sa.tz II].
Lemma 1.51
(i) 'R IPP' = (P2 ® /) 'R'
(ii) IPI2 'R* = R*(P2 ® /) '
(iii) 'R*'R IPI2 = IPI2 'R*'R .
Proof. (i) By means of Theorem 1.46 and the relations (1.47) and (1.5) we derive
'R IPI2 (211')(q-l)/'l (F* ®/)V FIPI2 = (211')(H)/2 (F* 18) /) VIQI2 F
= (211' )<q-1)/2 ( F*Q2 ® /)V F = (211' ){q-1)/2 ( p2 F* ® I) V F
= (P2 ® /)'R.
(ii) The equality is obtained by taking adjoints in (i). (iii) The equality follows from (i) and (ii).
55
0
Now we can prove the Radon inversion formula, as established by John [31, (Ll4), (l.l5)].
Theorem l.li2 Let f E S(JR9). Then
{
fl.(q-t)/2'R,*'R/ for q odd , (i) I = i{21ri)H
fl.(w-2)/2'R,*(1tP®/)'R./ for qeven.
(ii) For q odd we have
f(x) = i(211'i)1-q fl.(H)/Z j ('RI) (x · w,w) duq- 1(w) , x E JR:I.
s•-' For q even we have
f(x) = (211'it" fl.(q- 2)12 j <f (p- (x ·w))-1 ((flfop)'R.f) (p,w)dp) du9-1(w),
sq-l R
xe/R9.
Proof. (i) The formula is obtained from Theorem 1..50 (i) and Lemmas 1.51 (ii) and 1.48 (ii). (ii) The integral expressions follow from assertion (i) and Lemmas 1.49 and 1.48 (iii). 0
As an application we consider the Cauchy problem for the wave equation in /Rq, with odd q;:::: 3;
{
tl.u = u11 , x E /R9 , t E IR ,
u(x;O) = ;p(x), u1(x;O) = 1/;(x}, x E /Rq, ( 1.84)
56
with initial values 1(), 'ljJ E C8"(/R.9). We solve the Cauchy problem {1.84) by means of the Radon transformation and we derive the strong Huygens' principle for the solution, cf .. Lax and Phillips [34, Chapter IV] and Petkov [43, Chapter 11].
Theorem 1.53 Let q ;::: 3 be odd and let 1(), 'ljJ E C8"(1m). Then the Cauchy problem ( 1. 84) has a unique solution given by
u(x; t) = ~(21ri) 1 -9 ~(q-1 )12 j ('RI()) (x · w- t,w) duH(w) sq-1
-!{21ri)1- 9 ~(q-3)/2 j ((8f8p)'R.,P) (x · w- t,w) du9- 1(w), {x; t) E Im x IR.
sq-1
Proof. The Cauchy problem {1.84) has a unique solution by [34, IV Theo~em 1.1]. We introduce the Radon transforms
u(p,w;t) = ('Ru( ·;t)) (p,w), {p,w) E Z9 , t E IR,
By applying the Radon transformation to (1.84) and by using Lemma 1.51 (i), we are led to the following Cauchy problem for the one-dimensional wave equation:
{
~PP = uu , .(p, w) E Z9 , t E IR ,
u(p,w;O) = l()(p,w), u1(p,w;O) = ~(p,w), (p,w) E Z9 •
The solution of this Cauchy problem is known to be
p+t
u(p,w;t)=h?(p+t,w)+~<,?(p-t,w)+~ j ~(s,w)ds, (p,w)EZ9 , teJR. p-t
By means of the Radon inversion formula of Theorem 1.52 and Lemma 1.51 (ii) we obtain the solution of the Cauchy problem (1.84),
u(x; t) = !{21ri)1-q [~(q- 1 )/2 'R*(~<,?(p + t,w) + ~<,?(p- t,w))
p+t
-~(q-3)/2'R*(P2 ®I) G j .J,(s,w)ds)] p-t
= H21rip-q ~(q- 1 )12 j (<,?(x · w + t,w) + <,?(x · w- t,w)) du9-1 (w)
sq-1
+H21ri)1-q ~(q-3)/2 j (~"P(x · w + t,w)- ~p(x · w- t,w)) du9-1(w) ,
sq-1
(x;t) E JR9 x IR.
57
Since cp is even on Z9 and since .(fiP is odd on Z9 , we have
j cp(x · w + t,w) duH(w) = j cp(x · w- t,w) du9-1(w) ,
s•-• s•-•
j .(fip(x • w + t,w) du9-1(w) = - j .(fip(x · w- t,w) du9-1(w) . s•-• s•-•
Combination of the latter three results yields the stated solution/for u(x; t). 0
Now we can give a short proof of the strong Huygens' principle, cf. [43, Corollary 2.3.4].
Corollary 1.54 (strong Huygens' principle) Let q ?: 3 be odd, let B > 0 and let cp,t/J E C0 (/R9 ) such that cp(x) = t/J(x) = 0, lxl?: B. Then for ltl > B, the solution u(x;t) of the Cauchy problem {1.84) vanishes for lxl < lti-B.
Proof. Let (x;t) E /R9 x 1R such that itl > Band lxl < ltl- B. Then for wE S9-1
one has
lx · w- tl ?: ltl - lx · wl ?: ltl- lxl > B ,
and consequently,
('Rep) (x · w- t,w) = 0, ((ofop) 'Rt/J) (x · w- t,w) = o . Thus the solution u(x; t), as given in Theorem 1.53, vanishes. 0
In the n~xt theorem we characterize the range of the Radon transformation when applied to the radially symmetric functions in S(/R9 ). In this characterization the Erdelyi-Kober operators lW.\, A E JR, defined in (1.23), appear. We recall that the operators lW.\, A E /R, constitute a group of continuous operators on Seven(IR), see (1.2.')).
'l:heorem 1.55 (i) Let foE Seven(IR) and define the radially symmetric function f: /R9 -+ C by
f(rw) = fo(r), r?: 0, wE S9-1
•
Then f E S(/R9 ) and its Radon transform is given by
(RI) (p,w) = (21r)(q-t)/2(1W(9-t)/2fo) (p)
The Erdelyi-I<ober transform IW(q-1)/2 fo belongs to Seven( IR).
(ii) Let go E Seven(IR) and define the function g : Z9 -+ C by
g(p,w) = go(p), (p,w) E Z9 •
58
Then g is the Radon transform of the function n.-1g given by
('R--1g) (rw) = (211")-(q-1)/2 (IW-(q-1}/2go) (r)
{
(211")-(q-1)/2 (-r-1d/dr)<9- 1l/2 g0 (r) for q odd,
= 2(211")-q/2 (-r-1d/dr)912 i g0 (p) (p2 - r2)-112 pdp for q even,
r 2::0, wE S9- 1 •
The Erdelyi-Kober transform IW:...(q-1)/2go belongs to Seven(IR).
Proof. (i) According to Theorem 1.36 we have f E S(/R9). By means of Theorem 1.46 and 1.33 (Hecke-Bochner) and the relations (1.24) and (1.23) we deduce for p 2:: 0, w E S9- 1,
(211" t<q-1)/2 ('RI) (p, w) = ((IF* 18> I) V IF f) (p, w) = (IH -1/2/Hq/2-tfo) (~)
As observed, we have IW(q-1}/2 /o E Seven(/R) by (1.25). (ii) The assertion readily follows by inversion of the result from assertion (i). 0
At the end of this section we want to characterize the range of the Radon transformation when applied to functions f E S(JR9) of the form
J(rw) = /m(r) Ym(w) with /mE Qm(Seven(/R)), Ym E Y:!,.
In this characterization Weyl-Gegenbauer transformations are involved. In Chapter 3 we study these transformations at length. Here we need their definition and some of their integral representations only; see (3.37) and Theorems 3.22 and 3.38. Let m, q E /N0 , q ;::: 2. The Weyl-Gegenbauer transformation W9m,q/2_1 is defined on the space S_(JR+) = {! E C00(JR+): x" j<1> E Loo([l, oo)) , k, lE /N0 } by
(1.85)
Here the operators WA, A E /R, are the Weyl operators defined in (1.34), while the operators IWA, A E /R, are the Erdelyi-Kober operators defined in (1.23). The operator W9m,q/2 _ 1 is a bijection on S .... (JR+) with inverse
WQ;;-,~q12_ 1 = QmlW_m-(q-1}/2Wm. (1.86)
Since the Erdelyi-Kober operator IWm+(q-1)/2 maps Seven(IR) onto itself, by (1.25), we have
(1.87)
In this connection we· refer to Theorem 1.6a where we have characterized the spaces qm(Seven(IR)) and pm(Seven(IR)). Let I E S .... (JR+). Both the function W9m,q/2-l I and the function wg;;:.~q/2-1 I have an integral representation involving the Gegenbauer polynomial CU2- 1. By setting (cf. (3.32))
cq/2-1 = 2(3-q)/2 r(q- 2) m! cq/2-1 "' f(m+q-2}f(q/2-l/2) m
we have 00
(WQm,q/2-l f) (x) = I (t2 - x2}(q-S)/2 C!{2- 1(x/t) f(t) t dt '
00
(wg-1 /) ( ) (-1}'~-t I (t2-x2)(q-3)/2cqm/2-l(t/.x) /(q-ll(t)dt. m,q/2-l x = xq-2 \
Theorem 1.56 (i) Let f.,. e Q"'(Seven(B.)), let Ym e Y:. and define I: Hl' -+ c by
J(rw) = /m(r) Y.,.(w), r;?: 0, we 89- 1 .
Then f e S(JR'~) and its Radon transform is given by
('Rf) (p,w) = (21f}(q-l)/2 (WQm,q/2-1 /m) (p} Ym(w) 00
= (27r)(q-1)/2 I (r2- p2)(q-3)/2 c!{2-1(p/r) fm(1') 7' d!· Ym(w) ' ! p
P ~ o • w e sq-1 . The Weyl-Gegenbauer transform WQm,q/2-1 fm belongs to P"'(Seveu(IR.)).
(ii) Let g.,. e pm(Seven(IR)), let Ym e JJ:. and define g: Z9 -+ C by
g(p,w) = g.,.(p) Ym(w) , (p,w) e Z9 •
Then g is the Radon transform of the function 'R-1 g given by
('R-1g) (rw) = (21ft(q-l)/2 (WQ;;.~q/2_ 1 9m) (r) Ym(w)
= ( -J2Ji)I-q 1"" rq-2 (p2 - r2)(q-3)/2 c!{2-l(p/r) g~;-l>(p) dp Y,n(w) '
T
1· ~ o , w e sq-t . The inverse Weyl-Gegenbauer transform wg;;_~912_ 1 9m belongs to Q"'(S.,ven(IR.)).
Proof. (i) According to Theorem 1.37 we have f e S(JR:l).
59 .
(1.88)
(1.89)
(1.90)
By means of Theorems 1.46 and 1.33 (Hecke-Bochner) and relations (1..5) and (1.24) we deduce for p ;?: 0, w e sq-l,
(21ft(q-t)fz('Rf) (p,w) = ((JF*@ I) VJFJ) (p,w)
= (-i}m (JF*QmDfm+q/2-tQ-m fm) (p) Ym(w)
= im(pm JF* Dfm+q/2-lQ-m /m) (p) Ym(w)
(W-mDf-1/ZDfm+q/2-lQ-m /m) (p) Ym(w)
= (W-mlWm+(q-l)/2Q-m /m) (p) Ym(w) .
The proof is completed by using the relations (1.85), (1.89) and (1.87). (ii) The assertion readily follows by inversion of the result from assertion (i). D
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CHAPTER 2
Gel'fand-Shilov spaces
2.1 The Gel'fand-Shilov spaces S!(B)
61
In their celebrated treatise on generalized functions [22, Chapter IV] Gel'fand and Shilov introduced some new suhspaces of the Schwartz space S(JR'I), ~hich they called spaces of type S. In this section we gather some properties of these spaces for q = 1.
Definition 2.1 Let a, {J ~ 0. (i) The space S:'(.IR.) consists of all functions f E S(JR) for which their exist A, Bt > 0 (I E /N0), such that
llxk J{1liiL..,(R) 5 BtAk(k!)" , k, lE /No .
(ii) The space S!,(JR) consists of all functions f E S(/R) for which their exist Ak, B > 0 (k E .DV0 ), such that
llx"' f 11liiL..,(R) 5 A,.B1 (l!)P , k, l E .DVo .
(iii) The space se(JR) consists of all functions f E S(/R) for which their exist A, B, C > 0, such that
Because of the inequality e-mmm 5 m! 5 mm, cf. Lemma 2.4 (i), the factorials k! and l! in Definition 2.1 may he replaced by k"' and 11, respectively. Let JRt = JR+ U { 0}, JR't, = JR+ U { oo} and JRt.oo = JR+ U { 0, oo}. To make some of the subsequent results also valid for (a, {J) = (oo, oo), we define s:(IR) S(JR). We shall call the spaces se(JR), a,{3 E JRt.oo, (a,{J) ::f:. (oo,oo), Gel'fand-Shilov spaces. We observe that Gel'fa.nd and Shilov used the notations Sa, SiJ and sg for S';'(/R), S!(JR) and se(IR), respectively. We do not adopt their notation because our notation does not discriminate b~tween S,., SiJ and se, as a result of which many properties of the Gel'fand-Shilov spaces can be formulated concisely. The multiplication operator X and the differentiation operator djdx map se(JR) into itself. The reader may observe that the smaller the numbers a and {3, the stronger are the constraints on the functions in se(JR). So the question arises whether there exist numbers a and {3, such that the constraints are so strong that Sg(JR) becomes trivial, i.e. contains the null function only. In this connection we present the following theorem, see [22, Chapter IV, Sections 2.1, 8.1, 8.2].
Theorem 2.2 Let a, {3 E JR+. Then
(i) C(f'(/R) S(f'(/R) c S:'(IR) ;
(ii) /F(C(f'(/R)) = S~(JR) c S!,(IR);
(iii) S~(/R) is nontrivial if and only if o > 1 ;
62
(iv) sg(JR) is nontrivial if and only if fJ > f;
(v) S~(JR) is nontrivial if and only if a+ fJ ~ 1 .
For fJ $ 1, the functions in S~(JR) have an analytic continuation in the complex plane with a well-specified growth behaviour, see [22, Chapter IV, Sections 2.2, 2.3, 7, 8].
Theorem 2.3 Let a E JR.!,. Then (i) a function f E S!( JR) can be extended to an analytic function on the strip IYI < 1/ B of the complez plane z = x + iy; (ii) a function f E S~(/R) with fJ < 1 can be extended to an entire analytic function.
In the next two lemmas we present several estimations which we shall use frequently in the sequel.
Lemma 2.4 Let k, l E /N0 , ·let x, t, a E JR.+. Then
(i) e-kkk $ k! $ kk ;
(ii) (k + l)! $ 21<+1 k! /! ;
(iii) (k -1)! $ k!fl!, provided that k 2: l;
(iv) Hk/al !)" 5 (e2(1 + 1/a))k k!;
(v)
(vi)
x" exp( -tx11"') $ (kaf(et))ka 5 (aft)k"' (k!)'" ;
inf (x-mmm") 5 exp(ae/2) exp(-(a/e)x11"). mENo· ·
Proof. (i) e,. = E::":o knfk! 2: kkfk! 2: 1;
(ii) (k + l)! et') k! 11 s 21<+1 k! ,, ;
(iii) (k -I)!= k!/(l!m) $ k!fl!, provided that k;::: l; (iv) By means of assertion (i) we estimate
Uk/al!)"' $ rk/al"r~<Jal $ (kfa + t)"'<"la+t) = (kfa +It (kfa + t),.
$ e"(kfa + k)k = (e(l + 1/a))" kk $ (e2(1 + 1/o))k k!.
(v) On JR.+, the function f(x) = x" exp(-tx11<>) attains its maximum at x = (koft)"'. (vi) The inequality is taken from [22, p. 171]. 0
Lemma 2.5 Let f E S(/R). Then
(i) II/IIL2<Rl $ ft(li/IILoo(R) + llx/IILoo(B)) ;
(ii) 11/IILoo(R) 5 ~ (II/IIL2(R) + II/'IIL,(n));
63
mln(2k,l) ( k) (l) (iii) IIQ" P' IIILcR> $ [; ~ i j! IIQ2"-i lll~r.~cR> IIP21
-; lll~r.~IRJ , k, l E /No .
Proof. (i) IIIIIL(R) $;sup {(1 +x2) ll(x)IZ) j (1 +x2t 1 dx $; li'(IIIIILoo(R)+ llxfiiLoo(R))2
; o:ER R
(ii) ll(x)l$ !:= j l(lFI)(y)ldy$; ~(j(l+y2)1(1F/)(y)l2 dy) 1 12 v211' R v2 R
(iii) since the operators P and Q are self-adjoint we derive by means of Leibniz's differentiation rule for k, 1 E /No,
min(2k,l) (2k) (l) kl 2 l2kl .· . 21' 2fc' IIQ p lll~r.~tR) = (P Q pI, I)L2(R) = ~ ,, . . j! (P -J I, Q -J /)L2(R} . 1=0 J J
Now we obtain the w~nted result by the Cauchy-Schwarz inequality. 0
We give several characterizations of the Gel'fand-Shilov spaces. For the corresponding characterization of S( /R) we refer to Theorem 1.2.
Theorem 2.6 Let I E S(/R). I. Let a E ~. Then the following assertions are equivalent:
(i) I E S';'(/R) ;
(ii) there exist A,B, > 0 (lE /No) such that llxfci(I}IIL2(R} $ B,A"(k!)"', k,l E /No;
(iii) there exist A, B > 0 such that llx" lll~r.~cRJ $; BAfc(k!)"' , k E /N0 ;
(iv)there exist A,B > 0 such that llx"IIILoo(R} $; BA"(k!)", k E /No.
II. Let {J E ~. Then the following assertions are equivalent:
(i) I E S!,(JR) ;
{ii) there exist Ak, B > 0 (k E /No) such that llxfc l('liiLl(R} :5 AkB1(l!)i3 , k, lE /No ;
{iii) there exist A, B > 0 such that llf('}IIL2 (R) $; AB1{1!)!3 , lE /No ;
(iv) there exist A, B > 0 such that llf<'lll£ooiR) $; AB1(l!)i3, lE /N0 •
Ill. Let a, {J E ~ with a + {J 2:: 1. Then the following asstTlions are equivalent:
(i) I E S~(/R) ;
(ii) there exist A,B,C > 0 such that llxfcl!'liiL21Rl $; CAkB1(k!)"(l!)i3, k,l E /No;
64
(iii) there exist A, B > 0 such that
(iv)-there exist A, B > 0 such that
Proof. I. We follow the scheme (i) :::} (iv) => (iii) => (ii) => (i). Obviously (i):::} (iv). For k E /N0 we have
IIQk/lll.(R) = (Q2kf,f)~(R) :5llx2k/IIL.,.(R) II/IIL1(R) · Via this inequality and Lemma 2.4 (ii), it follows that (iv) => (iii). Suppose f satisfies (iii). By means of Lemma 2.5 (iii) and Lemma 2.4 (ii), we estimate for k,l E /No,
min(2k,l) (2k) (') . IIQk pl /lli,(R) :5 L: . . j! BA2k-j((2:k- j)!)" IIP2l-j /IIL.(R)
)..0 J J
That is, f satisfies (ii). By means of Lemma 2.5 (ii) it follows that (ii) => (i).
11. We follow the scheme (i) :::} (iv) :::} (iii) :::} (ii) :::} (i). Obviously (i) :::} (iv). For lE /No we have
IIP1flli.cR> = (P21f,f)L.(RJ :5IIJ<21JIIL.,.(RJ II/IIL1(RJ .
Via this inequality and Lemma 2.4 (ii), it follows that (iv) :::} (iii). Suppose f satisfies (iii). By means of Lemma 2.5 (iii) and Lemma 2.4 (ii), we estimate for k,l E /N0 ,
That is, f satisfies (ii). By means of Lemma 2.5 (ii) it follows that (ii) => (i).
Ill. We follow the scheme (i) :::} (iv) :::} (iii) :::} (ii) :::} (i). Obviously (i) :::} {iv). From the implication (iv) :::} (iii) in I and in II we obtain (iv) :::} (iii). Suppose f satisfies (iii). By means of Lemma 2.5 (iii) and Lemma 2.4 (ii), (iii) and by the assumption that a + /3 ~ I, we estimate for k, l E JN0 ,
65
min(2k,1) (2k) (') IIQ/cPI/IIL(B) ~ ~ . . j! BA2"-i((2k- j)!)0 AB21-j((2l- j)!)~' J=O ) 1
~AB( A+ 1)2" (B + 1)21 ((2k)!)" ((21)!)~' ~ 2. . (j!)Ha+Pl mln(2k,l) ( k) (')
J=O 1 )
That is, f satisfies (ii). By means of Lemma 2.5 (ii) it follows that (ii) => (i). 0
From Theorem 2.6 we immediately obtain for a + (3 ~ 1,
8:'(/R) n S~{IR) = S~(IR). (2.1)
This intersection result wa.s first proved by Ka.shpirovskii [33]. He states that this result is also true for a+ (3 < 1: "In the case when a+ (3 < 1 (S! consists of the unique function f(x) = 0), Eq. (2.1) follows from the Phragmen-Lindelof and Liouville theorems." Since this argument is not quite obvious, we present another proof here which is inspired by Duran's paper [11] ..
Theorem 2. 7 Let a, (3 E JRd. Then
S:'(IR) n S!,(JR) S!(JR) .
Proof. As observed, only the case 0 ~ a + (3 < 1 remains to be considered. Let f E S:'(IR) n S!(JR). By Theorem 2.6 there exist A, B > 0 such that
IIQ" fiiL2(B) ~ BAk(k!)", IIP1fiiL2(B) ~ AB1(1!)19 , k, lE /No.
Hence there exist C, D > 0 such that
IIQmPkfiiL(B) ~ CD2k(k!)l+a+P, k E IN, mE {k,k -1}.
Indeed, by means of Lemma. 2.5 (iii) and Lemma 2.4 (ii), (iii) we estimate fork E IN, mE {k,k-1},
~ AB(2(A + 1) (B + l))2k {{2k)!)"+P ~ G) (j!) 1-("+fl)
~ AB(2(A + 1) (B + 1))2k (22k(k!)2)'"+P ~ G) (k!)H<>+P)
66
Next we derive fork E IN, by means of Lemma 2.5 (ii),
IIQ"' P"' /IIL .. (R) ::::; ~ (I IQ"' pk /II~(R) + k IIQ"'-l pk fii~(R) + IIQ" pk+l /II~(R})
5 ~ D"'(k!)(l+,.+PI/2 (1 + k + D(k + l)(l+,.+Pl/2 )
5 ~ D"'(k!)(t+a+Pl/2 (D + l)(k + 1) $; ~ (D + 1){2D)"' (k!)(l+,.+Pl/:1.
Since/ E S!o(JR) with fJ < 1, the function f can be extended to an entire analytic function, by Theorem 2.3 (ii). Let z E C. Then there exist x ~ 0 and 0 ::::; t/J ::::; 211" such that
z = x+2xe;".
By applying the L00-estimate for Q"' P"' /,we deduce 00 00
lf(z)l = lf(x + 2xei"')l = I E J<"'>(x) (k!)-1 (2xei")kl ::::; E 2-"(k!)-1 lxk /(kl(x)l k~ k~
5 11/IIL..,(R) + !!_ (D + 1) f: (4D)" (k!)<a+P-tl/2 • . .../2 k=l .
Hence f is bounded on C. Then f must be constant, due to Li~ville's theorem. As f(x)-+ 0 (lxl-+ oo) we have /(x) = 0, x E 11. Thus we have shown that S;:>(B) n S!o(B) is the trivial space. Since, by Definition 2.1, S~(B) C S;:>(B) n S!o(B), we are done. In this manner we have also proved that S~(B) is trivial if 0 ::::; a + ,9 < 1. 0
The action of the Fourier transformation on a Gel'fand-Shilov space swaps the indices a and {J, while the operators P and Q map S~(B) continuously onto itself; compare Theorem 1.4.
Theorem 2.8 Let a,{J E JR(t,00 • Then
(i) /F(S~(B)) = Sp(B) ,
(ii) Qn(S~(B))={JeS~(B) J<"'l(O)=O, k=O, ... ,n-1}, nEIN,
(iii) pn(S~(B))={fES~(JR) j f(y)y"'dy=O, k O, ... ,n-1}, neJN. R
Proof. (i) For I E S(B) we have
IIQ"' pl IF III~(R) = 11 pkQ/ fiiL.(R) , k, l E /No ·
By using this equality with l = 0 or k = 0, the result follows from Theorem 2.6. (ii) Let n E IN and let I E S~( 11) with l(ki(O) = 0 , k = 0, ... , n- 1. According to the proof of Theorem 1.4 (ii), there exists a function g E S(B) such that g(x) xn l(x) and
67
11 (l)ll < I! 11/(n+l)ll I E JN,o • 9 Loo(R) - (n + l)! Loo(R) ,
By Theorem 2.6, it readily follows that g E S!(JR) and therefore, f E Q"(S!(JR)). Conversely, suppose f E Q"(Sg(JR)). Then obviously f E S!(JR) with Jl"l(O) = 0 , k = 0, ... ,n -1. (iii) The proof runs along the same lines as the proof of Theorem 1.4 (iii). o
Corollary 2.9 Let a, {J E JRi.oo and let f : .IR -+ C. Then the following assertions are equivalent:
(i) I E S~(.IR) ;
(ii) / E S:'(.IR) and IF/ E Sp>(.IR).
Proof. The equivalence is a straightforward consequence of Theorems 2.7 and 2.8 (i). 0
We are going to relate the Gel'fand-Shilov spaces to Gevrey spaces brought about by the operators P and Q. First we introduce some notations taken from Ter Elst's thesis [17]. Let T be a self-adjoint operator in a Hilbert space H. For ,\ ;::: 0 and A > 0 the normed space S.\;A(T) is defined by
S-';A(T) = {/ E D00 (T) : 11/llr,-'I;A =sup (IIT"/IIu A-"(n!t-') < oo}.' (2.2) nE No
The Gevrey space S.\(T), brought about by T, is taken to be the union
S.\(T) u S>.;A(T) . (2.3) A>O
Clearly S;~.(T) C S~'(T) if,\ < p. The topology for S>.(T) is the inductive limit topology generated by the normed spaces S.\;A(T), A > 0. Since the topology of S>.(T) is regular, see [17, Theorem 1.11], we obtain the following characterization of sequential convergence in S;~.(T). For a sequence (/~c)kEN C D00(T) we have
/~:-+ 0 (k-+ oo) in S;.(T) if and only if there exists A > 0 such that
/~c E S;~.;A(T), k E lN, and 11/Jcllr,;~.;A-+ 0 (k-+ oo).
Consistent with (2.2) and (2.3), we define Soo(T) D""(T). For a unitary operator U on H we have, cf. (1.3),
(2.4)
S;~.(UTU*) = U(S;..(T)), ..\ E JRt.oo. (2 . .5)
In particular we have for the Gevrey spaces S >. ( P) and S,, ( Q ), brought about by the operators P and Q, respectively, via the Fourier transformation,
S>.(P) = IF(S;..(Q)), S;..(Q) = IF(S;..(P)), ,\ E JRt.oo, (2.6)
' cf. (1.6). The space S).(Q), A E JR+, admits the following characterization.
Lemma 2.10 Let~ E JR+. Then
. S).(Q) = {/ E L2(JR) : 3t>P exp(tjxj1/).)/ E Lz(lR)}.
Proof. Let J E S>.(Q). Then there exist A,B > 0 such that
IIQk /IILcRl S BAk(k!)2>. , k e No .
Let t = (4e2(1 + 1/(2~)) (A+ 1)1/(:1>.))-1• Then, by means of Lemma. 2.4 (iv) we have
I . .,., (2t)k I I exp(t lzl1/>.) /(z)l2 dx = E k! lxlkf>. l/(x)l2 d.x .R lo=O ' .R
S J;~ (2;t (11Qfk/(2All /lli2cRl + 11/IILcR))
s Be2' + f: (2tr BAfk/(2.\)1 Uk/(2~>1'>2" k=O k.
\
$ Be21 + E (2tt B(A + l)kf(2>.)+I (e2(1 + I/{2~}))k k! k=O k.
00
= Be21 + B( A + 1) E 2-" < oo . k=O
Hence, exp(t lxl 11-') f E L2(1R) fort as indicated. For the converse, let J E L2( JR) and suppose there exists t > 0 such that
exp(t lxjii.\) f E L2(JR) .
Then for k E 1'/0 we have
j lxk f(xW dx S sup lx" exp( -t lxl1/>.)l 2 I I exp(t lxl1'") /(x)l2 dx
.R MR .R
$ (~/t)2AI: (k!)2.\ I I exp(t lxl1/>.) /(z)l2 dx .R
where we used Lemma 2.4 (v) for the latter inequality. Hence, f E S>.(Q)~ 0
In Theorem 1.3 we have seen that S(JR) = Soo(Q) n S,x,(P). The Gel'fand-Shilov spaces admit a similar functional analytic characterization in terms of Gevrey spaces, cf. Van Eijndhoven [12, Theorem 4.5].
Theorem 2.11 Let a,/1 E JR.t,w Then
S~(JR) = Sa(Q) n Sp(P) .
Proof. For a E JRt we have, by Theorem 2.6 I and Theorem 1.3,
S:'(JR) = s .. (Q) n S(.R) = s .. (Q) n Soo(P).
For {3 E JRt we have, by Theorem 2.8 (i), the previous result and relation (2.6),
S!(.R) = /F(Bp(.R)) = JF(Sp(Q)) n F(Soo(P)) = Sp(P) n S00(Q) .
For a,/3 E JRt we have, by Theorem 2.7 and the previous two results,
S~(.R) = S;:>(.R) n S!,(.R) = (S,.(Q) n Soo(P)) n (S13(P) n Soo(Q))
= Sa(Q) n Sp(P) .
69
0
Remarks. (1) The equality in Theorem 2.11 was first proved by Van Eijndhoven [12, Theorem 4.5] in the case a + {3 ~ 1 with a, {3 E JR.t,, and later by Ter Elst [17, Theorem 3.2] in the case a + {3 ~ 1 with a, /3 E JRt. (2) Since the Gel'fand-Shilov space Sf(.R) can thus be written as the intersection of the well-defined topological spaces Sa(Q) and Sp(P), we endow S~(IR) with the intersection topology. It, can be shown that sequential convergence in the intersection topology of sg(.R) agrees with Gel'fand and Shilov's definition of sequential convergence in sg(JR), cf. [22, Chapter IV, Section 3]. The space sg(JR) is complete. (3) The operators P and Q map S~(JR) continuously into itself. The operator IF maps S~(JR.) continuously onto Sp(JR).
The next theorem states that functions in S;:>(JR) decrease exponentially at infinity.
Theorem 2.12 Let a E JR.+ and let f E S(JR.). Then thefollowing assertions are equivalent:
(i) f E S;:>(JR.) ;
(ii) there exists t > 0 such that exp(t lxl*) f E L2(JR.);
(iii) there exists t > 0 such that exp(t lxl11") f E Loo( JR.) .
Proof. By Lemma 2.10 and Theorem 2.11 we have (i) # (ii). Obviously, (iii) => (ii). We show that (i) => (iii). Suppose f E S;:>(JR). By Definition 2.1 (i) there exist A,C > 0 such that
lf(x)l :5 C(lxi/A)-k kk", x E JR., k E /No.
Hence, by Lemma 2.4 (vi),
lf(x)l :5 C k~nJ0 ((lxi/A)-k kka) :5 C exp(ae/2) exp(-(a/e) (lxi/A)ll"'), x E JR..
0
The characterizations of S:'(JR.) established in Theorems 2.6 I and 2.12 lead, by Corollary 2.9, to analogous characterizations of S!(JR.) and S~(JR), a, /3 E JR+.
70
Recalling the characterization of S(.Ul.) by means of the Fourier transformation in Theorem 1.5, we obtain the following characterizations of the Gel'fand-Shilov spaces.
Theorem 2.13 Let f E L2(.Ul.) and let p E {2, oo }. I. Let a E .m.+. Then the following assertions are equivalent:
(i) f E B:'(.Ul.) i
(ii) there exist A, B > 0 such that
llx.l:/llr,.(RJ $; BAk(k!)"' and x11Ff E Lp(.Ul.), k,l E /N0 ;
(iii) there exists t > 0 such that
exp(t lxl1'"') f E Lp(.Ul.) and x 11F f E Lp(.Ul.) , lE /No .
II. Let fJ E .m+. Then the following assertions are equivalent:
(i) f E S~(.Ul.) ;
(ii) there exist A, B > 0 such that
(iii) there exists t > 0 such that
xk f E Lp(.Ul.) and exp(t lxl 1'~') IFJ E L,(.Ul.), k E /No.
III. Let a, fJ E JR+. Then the following assertions are equivalent:
(i) f E S~(/R) ;
(ii) there exist A, B > 0 such that
(iii) there exists t > 0 such that
exp(t lxPia) f E L,(IR) and exp(t lxi1/.B) IF f E Lp( IR) .
As we have seen in Theorem 1.6, the functions in S(/R) can be characterized in terms of their Hermite expansion coefficients:
S(.Ul.) = D00(Pz + Q2) = {f E Lz(/R) : (n"(/,1/J,.)Lz(R)) E loo, k E !No}·
Here {1/J,. : n E /N0} denotes the Hermite basis in L2(/R) as introduced in (1.7). Zhang has proved the analogue for the functions in S~(/R) with o ~ &, see [51, Theorem 2 and Lemma3]:
Let a~ ! and let I E L2(JR.). Then I E s:(JR.) if and only if
3t>O (/, ~ .. )~(R) = O(exp( -tnl/('l<>))), n-+ oo .
71
This characterization of s:(JR.) is contained in the following theorem, which we provide with a new proof. First we present an auxiliary result.
Lemma 2.14 Let,\ E Bt. Then
Proof. Let I E L2(JR.) and let a,. = !(!, ~ .. )~(R)I• n E /No. Suppose I E S>.(P2 + Q2). Then there exist A, B > 0 such tba.t
00 00
E n2ka! :5 E (2n + 1)2k a!= II(P2 + Q2)k llll,(R) :5 BAk(k!)2>. ' k E /No . n=O n=O
Let t = (4e2 (1 + 1/(2,\)) (A+ t)l/(2>.))-1• Then, by means of Lemma 2.4 (iv) we have
oo oo (2tntf>.)" sup lexp(tn1/>.)a,.l2 :5 E E 1 a!
n€No n=O k=O k.
Hence, exp(tn1f>.) a .. E loo fort as indicated. For the converse, suppose there exists t > 0 such that exp( tn I/>.) a,. E l.., . Let 0 < T < t. Then exp(m1/>.)a,. E l2. Fork E /No we have
00 00
E (2n + 1 )2" a! :5 sup l(2n + 1)" exp{~m11>.)1 2 E I exp(m11>.) a,.l2
n=O ~~ n=O
00
$ (1 +3(.A/T)>.)2k(k!)2>. E lexp(ml/>.)anl2 n=O
where we used Lemma 2.4 (v) for the latter inequality. Hence, I E .':h,(P2 + Q2 ). 0
We arrive at the announced characterization of the space 8~(81), a;::::~:
Theorem 2.15 Let a :2: t· Then
72
The equalities hold as topological vector spaces.
(ii) For/ E S;(JR) we have
00
'/(x) = E (/,t/J,.)L,(R.) t/J,.(x), x E lR, n=O
where the series converges in s;(JR).
Proof. (i) The first and the third equality have been established in Theorem 2.11 and Lemma 2.14, respectively. We shall show that
s:(JR) c S2a(P2 + Q2)
and that
{/ E L2(JR) : 3t>O (exp(tn1/(
2"'l) (/,t/Jn)L,(R.)) E loo} C s:(JR).
Let f E S;(JR). For nE liV let V,. denote the set of operators
Qkt pit ... Q~" P" , k, l E N:; with lkl + Ill = n .
According to [12, Theorem 4.3 (iii)], there exist A, B > 0 such tha1\ for n E liV and Tn E V,.,
liT,. fiiL2<RI ::; BA"(n!)a .
Let n E liV. There exist operators T;,2,. E l/2,. , j = 1, ... , 2", such that
2"
(P2 + Q2t f = E Tj,2nf • i=l
Therefore, by the preceding estimate we have
II(P2 + Q2t /IIL2 (R.)::; 2"BA2"((2n)!t::; B(22"+1A2t (n!)2a.
That is,/ E S2,.(P2 + Q2).
Next, let f E L2(JR) and let there exist t > 0 such that (exp(tn11(2<>l) (/, t/Jn)L,(R.)) E 100 •
Then f E S(JR) by Theorem 1.6. By means of (1.10) we estimate fork E N 0 ; x E JR, 00
lx"f(x)l = 1 E (f,t/J,.)14<R.> x"t/J,.(x)l n=O
00
::; sup I exp(tnl/(2")) (/, tf.>n)L,(R)I 2"/2 E ((n. + k)!/n!)112 exp( -tnl/(2")) • nENo n=O
By using Lemma 2.4, we estimate
00
E ((n+k)!/n!)112 exp(-tn11{2al) n=O
k 00
::; E ((2k)!/k!)t/2+ E (n+k)k/2 exp(-tnl/(2a)) n=O n=k+l
73
00
~ (k+l) (22kk!)t/2+ .E (2n)k/2 exp(-tnl/(2a)) n=k+l
n=l
~ 22k+t(k!)" + 2,."(71'2 /6) (a/t)a(k+4) ((k + 4)!)"
~ 22k+l(k!)" + (71'2/6) (4!)" (2a/t)4" (4a/t)"k (k!)".
By combining the previous results, we infer that there exist A, B > 0 such that
llxk IIILoo(R) ~ BA,.(k!)" , k E /No .
That is, I E S;:'(JR), by Theorem 2.6 I (iv). Since I E L2(/R) and (exp(tnl/(2<>)) (!, t/Jn)L2(R)) E loo we also have IF I E L2(/R) and by (1.9), (exp(tn11(2<>l) (IF I, t/Jn)L2(R)) E 100 • By replacing I by IF I in the preceding derivation, we conclude that also IF I E S;:'(JR). Hence, I E S~(IR) by Corollary 2.9. Now it is not difficult to see that the equalities hold as topological vector spaces.
(ii) Let I E S~(/R) 1and let IN = L-:;01 (!, tPn)L2(R) tPn· We show that I - IN -+ 0 (N-+ oo) in S2,.(P2 + Q2). By assertion (i), there exists t > 0 such that
sup I exp(tnt/(2<>)) (/, tPn)Ll(R)I = B < oo · nE No
For NE IN we estimate, by using Lemma 2.4 (ii), (v), 00
II(P2 + Q2)k (f- IN)IIi2 (R) = .E (2n + 1)21< 1(1, tPn)L2(R)I 2
n=N 00
~ B2 ,E exp( -2tn11(2a)) (3n)2k n=N
00
~ B2 321< sup (exp( -tn11(2al) n2k) ,E exp( -tn1112"l) nEN n=N
00
~ B2(12a/t)4ak (k!)4" .E exp( -tnt/(2a)) . n=N
Hence I- IN E S2a;(t2a/t)2o (P2 + Q2), NE IN and
Ill- INIIP2+Q2,2a;(l2a/t)2o-+ 0 (N-+ oo).
That is, by (2.4), I- IN-+ 0 (N-+ oo) in S2a(P2 + Q2).
Theorem 2.16 Let a E JR:t.oo· Then
S,.(Q) n S,.(P) = S2a(P2 + Q2) if and only if a~~.
0
Proof. For a ~ ! we have S,.(Q) n S,.(P) = S2,.(P2 + Q2), by Theorem 2.15 (i). For
a < ! the space S,.(Q) n Sa(P) is trivial, by Theorems 2.11 and 2.2 (v), but the space S2,.(P2 + Q2) contains the Hermite functions t/Jn, nE /N0 , by Ll•mma 2.14. 0
74
2.2 The subspaces S~,even(.R) and the Hankel transformation lElv
In this section we consider the suhspace S~.-(JR) of even functions in S~(JR.). The topology on S~.even(.R) is the induced topology. Likewise, S!,odd(/R) is the suhspace of odd functions in S~(JR). The set-up of this section is similar to that of Section 1.2 where we discussed the space Seven(IR.). Let us start with a straightforward consequence of Theorem 2.8 (cf. Theorem 1.6a).
Theorem 2.17 Let a,fJ E Rt.oo and let nE /N. Then
(i) /F{S~,even(/R)) Sp,even(/R) •
(ii) Q(S~.even(/R)) = S~.odd(JR) ,
Q2n(S!,even(R)) = {! E S~,even(/R) : j<2"l(O) = 0, k = 0, ... ,n- I},
Q2n+l(S~,even(/R)) = {/ E S!,odd(R) : JIZk+tl(O)= 0, k = 0,: .. , n -1} ,
(iii) P(S!,even(JR.)) = S!,odd(/R) 1
{:J J 2k ' p~n(S~,even(JR.))={/ESa,even(/R) f(y)y dy=O, k=O, ... ,n~l}, R
pzn+l(S~,evenCR)) {/ES~,odd(R) j f(y)y 2"+tdy=0, k=O, ... ,n..:..l}. R
In Theorem 1.8 we have characterized the functions in Seven(R) in terms of their expansion coefficients with respect to the Laguerre basis {/L~ : nE /N0 } in Xzv+t• as introduced in (I.I2): For v ~ -!,
Seven(IR.) = {f E X2v+t : (n"(f,/L~)x2~+1 ) E loo, k E /No}·
Starting from Theorem 2.15 and using the same techniques as in the proof of Theorem 1.8, we establish the analogous characterization of the functions in S!,even(/R) with a~!·
Theorem 2.18 Let a ~ l and let 11 ~ -l· Then
(i) S!,even(/R) == {/ E Xzv+t : 3t>O (exp(tnl/(Za)) (/, /L~)x2.,+ 1 ) E loo}·
(ii) For f E S!,even(/R) we have
00
f(x) = L: (!, /L~)x2.,+ 1 /L~(x) , x E JR.+, n=O
where the series converges in S!,even(IR).
Next we shall study the Hankel transformation Dl., and the Erdelyi-Kober operator JW.~. defined in (1.16) and (1.23), respectively, on s~.even(/R). As we have seen in Theorem 1.9 and in (1.25), these operators are bijections on Seven(IR.). By (I.I7) we have for f E Seven(R},
75
(Ff) (x} = (.ILt/21} (x}, x > 0.
So it follows from Corollary 2.9 that a function f belongs to s~ ... ven(JR) if and only if
f E s:,even(JR) and /lLt/2/ E Sf..,ven(JR).
Now the natural question arises whether in this characterization the role of D/_1/ 2 can be taken over by Ill, with arbitrary v ~ -1/2. In Theorem 2.20 an affirmative answer to this question is given. , As a consequence we can characterize the functions j e S~.eve/JR) in terms of conditions of decrease at +oo of the function f and of its Hankel transform Dl,f (see Theorem 2.21}. Moreover, it then follows in a straightforward manner that (see Theorem 2.22)
lW>.(S!,even(JR)} = S~,even(/R), JH.,(S~.even(JR)) S3,even(/R) ·
For..\= mE 7t and v + 1/2 nE JN0 , these identities can easily be verified:
Lemma 2.19 Let a,p e JRt.w Then
(i) lWm(S!,even(JR)) = S~,even(JR), mE 7t, I
(ii) H-l/2+n(Sg,even(/R)) = S$,even(JR), ne /No .
Proof. (i) By {1.23) and Theorem 2.17 we have
IW_t(S~,even(JR)) = ( -x-1 dfdx) (S~,even(/R)) = S~,even(1R) . The identity to be proved is now obvious, since the operators lW>., ..\ e /R, constitute a group of continuous operators on s .. ven(JR), by (1.25). (ii) Let n E /No. By using (1.24) and the equality IH:112 = I we have
/H-l/2+n(S~,even(JR)) = IW_n(H-t/2(S~,even(JR)}) • The result to be proved then follows from Theorem 2.17 (i) and assertion (i). 0
We establish a generalization of Corollary 2.9.
Theorem 2.20 Let a, p e JRt.00 , let v ~ -1/2 and let J : 1R -+ C. Then the following assertions are equivalent:
(i) f E sg,even(JR) i
(ii) J E s:,even(/R) and Hvf E Slf:even(/R) .
Proof. First we consider the case a = oo, {3 0. From Theorem 2.2 (i) we know that S'if.even( /R) = C'if.even( /R), the subs pace of even functions in C0 ( ./R). The operators lW>., ..\ E .IR, map Seven( /R) onto itself, by ( 1.25 ). Hence it follows immediately from the definition of lW>. in (1.23), that IW,1(..\ E /R) maps S!r:even(/R) onto itself. Then, by using (1.24) and Theorem 2.17 (i) we have
Dfv(S~even(/R)) = H-t/2(lWv+l/2(S~ven(/R))) = ~,even(/R) •
76
Since JH; = I, the theorem holds true for a =' oo, /3 = 0. Next we consider the case a = oo, /3 E JR+. Suppose f E se,,.,ven(JR). Choose nE IV such that -1/2 + n > v. By Lemma 2.19 (ii) we have 1H-t/2+nf E Sif..,ven(/R). Hence, by Theorem 2.12, there exists t > 0 such that
exp(txl/.0) 1H-1/2+nf E L2(/R+) ·
By applying Lemma 1.11 with 11 replaced by -1/2 +nand p. replaced by 11 and W(x) = exp(ltx11P), we obtain
(1 + x2)v-n-l/2 exp(~tx1 111 ) IH,J E L2(JR+) .
Hence
expHtx1111 ) Dl,f E L2(JR+) .
That is, Dl,J E Sif..,ven(/R) by Theorem 2.12. Conversely, suppose f E s:;,even(/R) = Seven(/R) and Dl,J E Sif..,ven(JR). By Theorem 2.12 there exists t > 0 such that
exp(tx11P) Dl,f E L2(JR+) .
By Lemma 1.11 with p. = -1/2 and W(x) = exp(~tx1 1.0) it follows that
(1 + x2r"-3'2 exp(ltx11P) Df-t/21 E L:z(JR+) .
Hence
exp(~tx1 111 ) Df-t/2! E L2(JR+).
That is, Df_1tzf E S't,even(/R) by Theorem 2.12. Now Theorem 2.17 (i) yields f E se,,even(JR). So the theorem holds true for a oo, /3 E JR+. In case a E IRJ,00 , /3 = oo, the result of the theorem is obvious in view of Theorem 1.9. In case a, fJ E JRt, the theorem follows from the preceding results for a = oo or fJ = oo and Kashpirovskii's intersection result in Theorem 2.7. D
Recalling the characterization of Seven(IR) by means of the Hankel transformation Dl, in Theorem 1.12, we present the following characterization of the subspaces of even functions in the Gel'fa.nd-Shilov spaces, obtainable by means of Theorems 2.20 and 2.13.
Theorem 2.21 Let 11 ~-!,let f E X2,+I and let p E {2, oo}. I. Let a E JR+. Then the following assertions are equivalent:
(i) f E S':.,ven(/R) ;
(ii) there exist A, B > 0 such that
(iii) there exists t > 0 such that
11. Let {J E JR:+. Then the following assertions are equivalent:
(i) I e Sf!.,even(B) ;
(ii) there exist A, B > 0 such that
x" I E Lp(Jl+) and llx11H.,/IIL,.(R+) $ AB1(l!)P , k, I E /No ;
(iii) there exists t > 0 such that
x"f E Lp(B+) and exp(tx11P)JH111 E L,(JR:+), k E /N0 •
Ill. Let a, fJ E JR:+. Then the following assertions are equivalent:
(i) 1 e s~ ... -(B);
(ii) there exist A, B > 0 such that
(iii) there exists t > 0 such that
exp(tx11")/E L,(JR:+) and exp(tx11P)JH.,je L11(JR:+).
Theorem 2.22 Let aJ3 E ~00 , let v ~ -1/2 and let..\ E JR. Then
(i) JH.,(S~,even(B)) = S$,.,ven(B),
(ii) W-'(S~,even(B)) = S~ ... -(B) .
77
Proof. (i) From Theorem 2.20 with a= oo it follows that Dlv{S'if.,ven(/R)) = S!,even(/R). Now the wanted identity follows from the equality JH; = I in (I .20} and Kashpirovskii's intersection result in Theorem 2. 7. (ii) The identity follows from assertion (i) and (1.24), viz. DI"Di,J = Wv-pf for f E Seven{B), p,ll ~ -1/2. 0
The study of the Hankcl transformation lHv on S~,even( /R) ha.<> been the subject of a number of papers. Pathak [42] states that lHv{S~,even(R}) = S'$,eveu(R) in case 0 < a,{J < 1. However, the part of his proof, that is based on contour integration in the complex plane, is incorrect. A corrected proof has been given by Van Eijndhoven and Kerkhof [16]. Van Eijndhoven and De Graaf [15] have proved that Dlv(S;;,.ven(/R)) s:;,.,ven(R) in case l/2 $ a $ 1, by using properties of the Laguerre polynomials. The paper [13] by Van Eijndhoven and Van Berkel, contains part of the results of Theorems 2.20 and 2.21. Finally, we present a functional analytic characterization of Sf.,even( Dl).
78
Corollary 2.23 Let a,{J E JRi.oo and let 11;;:: -!· Then
S! •• ven(JR) = Sza(Q2) n Szp(li .. Q2111v),
where Q is the self-adjoint operator defined on {I E X211+t : xl E X211+t} by Qf = xl.
Proof. First we consider the case {J = oo, a :f:. oo. Let I E X2v+t· Observe that I E S00(li,_Q2llv) if and only if ll,_f E S00{Q2). So we have to show that the following assertions are equivalent:
(1) I E s:,.,ven(JR);
{2) there exist A, B > 0 such that
llx2" lllx2v+l :S BA"(k!)2
"' and x 21 llvl E Xzv+l , k, lE /No.
Suppose f satisfies (2). Then f E Seven(IR) by Corollary 1.13 and
llx2"111L(R+) :S 11/lli,.,(R+) + llx2" lllkzv+t • k E /No ·
Hence it follows that I E s:,e~(JR) by Theorem 2.6 I (iii). The converse, (1) => (2), follows from the definition of S':'(IR) and Theorem 1.9. Next we consider the case a= oo, {J :f:. oo. By observing that S2"'(li .. Q2 1H .. ) = II .. (S2.,(Q2
)),
"'f E JRi.oo and that 11?, = I, we deduce by means of Theorem 2.22 (i) and the previous result,
Sf!o,even(JR) = li.,(S!f:even(JR)) = li.,(S2p(Q2) n Soo(IH.,Q2 DI.,))
S2p(lllvQ2 IH.,) n Soo(Q2).
Finally, for a :/; oo and {J :/; oo the wanted identity follows from the previous results and Kashpirovskii's intersection result in Theorem 2.7. 0
2.3 'The spaces G~Cil~+) and the Hankel-Clifford transformation 1fv
In Theorem 1.15 (ii) we have shown that S(JR+) is equal to the space
{g E C00 (/R+) : the function X 1-t g(x2 ) belongs to Seven(IR)} .
The present section deals with the spaces G~(JR+) defined as follows:
Definition 2.24 Let a, {J E JRt.oo. Then
G~(JR+) = {g E C00 (/R+) : the function X 1-t g(x2) belongs to S!,even(JR)} .
The spaces G~(.Ji+) are related to the spaces G01 , GP and G~, introduced by Duran [llj. In [11, Definitions 2.1, 2.2 and 2.3], the latter spaces are defined by means of certain conditions on the norms
79
llx(k+I)/Zg(I)IIL.J(R+), k,l E /No, llx"g(t)IIL.J(R+), 0 ~ k ~ l/2.
Furthermore, Duran shows in [11, Corollary 3.8] that a function g belongs to the space· 0 01 , GfJ or G~, if and only if the function x 1-+ g(x2) belongs to the space Sa/2,everu S~/! or S~~~.even• respectively. Thus we have the connection
Our treatment of the spaces G~(JR+) is reverse to Duran's. We start from Definition 2.24 and then we characterize the spaces G~(JR+) by means of the operators x and d/dx. As a consequence, we deduce new characterizations of the spaces S!,even(JR} by means of the operators x and x-1 djdx.
The results for S!(R) and S!,even(JR) established in Sections 2.1 and 2.2 can be translated into results for G~(m+). We recall the definition of the Hankel-Ciifford transformation 1f.., in (1.27) and of the Weyl operator WA in (1.34). We also recall the formulas
from (1.29) and (1.35}. By means of Theorem 1.15 (ii) and various results from Sections 2.1 and 2.2 we gather the following properties of the spaces ae(m+).
Theorem 2.25 Let a,/3 E JRri,00 • Then
(ii) G~(R+) is nontrivial if and only if S!(JR) is nontrivial;
(iii) for {3 ~ 1, the functions in G~(R+) have an analytic c.ontinuation in the complex plane;
(v) let V~ -1/2 and let g: m+-+ C, then ' g e G~(m+) if and only if g e G';:'(m+) and 1t.,g e G';(m+) ;
(vi) rt.,(G~(m+)) = G<p(m+), v ~ -1/2;
(vii) WA(G~(Ol+)) = G~(R+) , .X E lR;
(viii) ae(JR+) = Sza(Q) n Szp(1t.,Q1t.,)' V~ 1/2 ' where Q is the self-adjoint operator defined on {g E X., : xg E X.,} by Qg = xg ;
(ix) G~(m+) = {g EX" : 31>o (exp(tni/(Za)) (g,.C~)x.) E loo} , a~ 1/2, v ~ -l, and forgE G~(R+) we have
00
g(x) = 2:: (g,.C~)x • .c:(x)' X Em+' n=O
where the series converges in G~(R+).
80
Next we shall derive several characterizations of the spaces G~(JR+) by means of the operators x and dfdx. We start with a characterization of G:'(JR+); by applying the HankelClifford transformation 'Ho, we obtain a. characterization of G!(JR+); then by means of the intersection result G:'(JR+) n G!(JR+) = G~(JR+) we arrive at a characterization of G~(:JR+).
Theorem 2.26 Let a,{J E IR; and let g E C'""(.fl+). Then
(i) g E G';'(JR+) if and only if there ezist A, B > 0 such that
llxk/2giiL2(R+) $ BAk(k!)" , x112g!l) E L2(JR+) , k, I E /No ;
(ii) g E G!(JR+) if and only if there ezist A,B > 0 such that
x"l2g e L2(JR+), llx112g(ll11L2!R+l $ AB1(l!)P, k,l e /No;
(iii)g E G~(JR+) if and only if there exist A, B > 0 such that
llxk/2giiL2(R+) $ BA"(k!)" , llx112g{lliiL2(.R+) $ AB1(l!) 13 , k, lE /No .
Proof. (i) The equivalence is a straightforward consequence of Theorems i.20 and 2.6 I. · (ii) By Theorem 2.25 (v) we have
g E G!,(JR+) if and only if 'H0g E G';(JR+) .
Since 'Ht is unitary on X~t lE /N0 , we have for hE S(JR+),
llx112'HohiiL2(R+) = II'Ht'Hohllx, =?! llx112h'1liiL2(R+) , lE /No .
Noting that 'H~ =I, we obtain the wanted equivalence from (i). (iii) The equivalence is obtained from (i) and (ii) because of Theorem 2.25 (iv). 0
In the above characterizations we want (1) to replace L2(JR+) and the L2(JR+)-norm by L00 (JR+) and the Loo(JR+)-norm; (2) to replace the conditions on xkl2g and x 112g<1> by conditions on x!k+l)/2g(ll; cf. the characterizations of S(JR+) in Theorem 1.20. As a preliminary we establish the following lemma.
Lemma 2.27 Let g E S(JR+). Then
(i) llx(k+l)/2g(I)IIL2(.R+) = 2k-l llx(k+I)/2('Hog)!k) IIL2(.R+) ' k, l E /No ;
(ii) llx(k+l)/2g(')lll2(R+) $ ml~,l) (miny,l)) (k; l) j!
Proof. (i) By means of (1.35), viz. 'Hp'Hv = 2~'-vWv-p, and the property that 'Hk+l is a unitary operator on xk+/, we derive
llx(k+l)/2g(lliiL2(R+) = 2-'II'H,'Hoollx.+,
= 2-'II'Hk+I'HI'Ho9llx.+, = 2k-l llx(k+l)/2 ('Hog)<kliiL2(R+) 0
(ii) First suppose k ~ lo Then, through integration by parts we have
00
llx(k+/)f2g<'lii~(R+) = j xk+lg(')(x)g<1>(x) dx 0
Next suppose k < lo Then, by (i) and the previous estimate,
llx(k+l)/29(l)IIL2(R+) = 22k-2/ llx(k+/)f2 ('Hog)<k)IIL2(R+)
Theorem 2.28 Let g E C""(JR.+)o I. Let a E JRt o Then the following assertions are equivalent:
(i) g E G';(JR.+) ;
(ii) there exist A,B, > 0 (lE /N0 ) such that
(iii) there exist A, B, > 0 (I E /N0 ) such that
(iv) there exist A, B > 0 such that
( v) there exist A, B > 0 such that
81
0
82
11. Let /3 E JRt. Then the following assertions are equivalent:
(i) g E G~(JR+) ;
(ii) there ezist A~c, B > 0 (k E /N0 ) such that
(iii) there ezist A~c, B > 0 (k E /N0 ) such that
(iv) there exist A, B > 0 such that
(v) there exist A, B > 0 suc,h that
Il I. Let a, /3 E JRt with a + /3 ~ 1. Then the following assertions are equivalent:
(i) g E G~(JR+) ;
(ii) there ezist A,B,G > 0 such that
(iii) there ezist A, B, G > 0 such that
(iv) there exist A, B > 0 such that
( v) there exist A, B > 0 such that
Proof. In each of the cases I, 11 and III, we follow the scheme (i) {:::} (iv), (ii) => (v) :::} (iv) :::} (iii) :::} (ii). The equivalence (i) {:::} (iv) is taken from Theorem 2.26. Obviously (ii) :::} (v). By using Theorem 1.20 it follows that (v) => (iv) in case I; in cases 11 and Ill the implication (v) => (iv) is obvious. By Lemma 2.27 (ii) we have (iv):::} (iii). Finally, the implication (iii) :::} (ii) follows by means of the following estimates for g E S(JR+),
a.nd
z
lx(A<+I}/2g(l)(xW = 2 Re f t(A<+I)/2g(')(t) (t(A<+I)/2g(ll(t))' dx
0
k E IV , l E lVo ,
00
2'+1 lx'l2g(')(x)l = 2lx'f2(1it'Ho9) (x)l =I j J,(..;Xy) ('Hog) (y)ytf2dyl 0
:::; 11(1 + Y) Y112'HooiiL2(.R+) :::; IIY112'Houii~(.R+) + 1Jy(I+Z)I2'Houii~(R+J
= 2'llu'12u<'1 11~<.R+J + 2'+211v('+2112u('+21 II~(.R+> , l e /No .
83
In the latter derivation we used the inequality IJ1(t)l =::; 1 fort~ 0, a.nd Lemma. 2.27 (i). 0
Remark. From Theorem 2.28 it follows that the condition
in [11, Definitions 2.1, 2.2 a.nd 2.3] is superfluous.
As a straightforward corollary of Theorem 2.28 we present the following characterizations of the spaces s~.evenUR).
Corollary 2.29 Let f E C""(JR+). I. Let a E JRri. Then the following assertions are equivalent:
(i) J E S::even(JR);
(ii) there exist A, B1 > 0 (l E /N0 ) such that
(iii) there exist A, B1 > 0 (I E IV0 ) such that
(iv) there exist A, B > 0 such that
(v) there exist A, B > 0 such that
11. Let fJ E JRt. Then the following assertiom are equivalent:
(i) I E s~,even(JR) ;
(ii) there exist A~o, B > 0 ( k E JN0 ) such that
Ux"+1(x-l d/dx)1/IIL .. (fi+)::; A~cW(l!)P, k,l E /No;
(iii)there exist A~c,B > 0 (k E /N0 ) such that
llx"+1(x-1 dfdx )i/ll~(fi+) $ A~cW(l!)11 , k, I E /No ;
(iv) there exist A, B > 0 such that
xlcf E L2(R.+), llx1(x-1 d/dx)1/llL2(1l:+)::; AB1(l!)P, k,l E /No;
(v) there exist A, B > 0 such that
xkJ E Loo( B.+) , 11(1 + x2 ) x1(x-1 d/dx)1/IIL .. (Ii:+J::; AB1(l!)P, k, lE /No.
Ill. Let a, {J E JRt with a+ {J ;?: 1. Then the following assertions are equivalent:
(i) 1 e s~..,_(B) ; (ii) there exist A; B, C > 0 such that
llxlc+1(x-1 d/dx)1f!iL .. cfi+J::; CAicB1(k!)"' (1!).0, k,l e /No;
(iii) there exist A, B, C > 0 such that
llxk+l(x-1 dfdx)1fll~cfi+) ~ CAkB1(kl)"' (1!).0, k,l E /No;
(iv) there exist A, B > 0 such that
llx~'fll~cfi+) ~ BA1'(k!)"', llx1(x-1 d/dx)1/ll~(fi+)::; AB1(l!)P, k, 1 E lNo;
( v} there exist A, B > 0 such that
k,l E /N0 •
2.4 The Gel'fand-Shilov spaces S~CIRq)
In this section we consider the Gel'fand-Shilov spaces S~(JR.q) and we gather some properties of these spaces for q;?: 2, cf. [22, Chapter IV, Section 9].
Definition 2.30 Let a, {J;?: 0. (i) The space S';'(Dlq) consists of all functions f E S(/R.q) for which there exist A, Bj > 0 (j E /No), such that ·
llx"d /IIL .. (fi•l ~ BttiAikl(lkl!)" , k, lE INJ .
85
(ii) The space S!,(JR'l) consists of all functions f E S(/R9 ) for which there exist A;, B > 0 (i E !No), such that
llxl:(i JIIL.,.(R~) $ AIA:IB1II(Ill!)11 , k,l E lNJ . (iii) The space se(JR'l) consists of all functions f E S(fR'l)forwhich there exist A,B,C > 0, such that
We recall the multi-index notations XI:= z~1 • ••• ·X~ and fl = a!• .... ·8!· with aj = a;axj, j = 1, ... ,q; furthermore, lkl = k1 + ... + k9 , k E INJ. To make some of the subsequent results also valid for (o:, /3) = (oo, oo), we define s:(fR'l) = S(JR'l). The multiplication operators Xj and the differentiation operators aj, j = 1, . .. ,q, map S~(JR9) into itself. The space se( JR9) is nontrivial if and only if se( JR) is non trivial, see [22, p. 243]. Later on in this section we will present several characterizations of S~( /Rq). For that purpose we introduce,some notations from Ter Elst's thesis [17]. Let T = (T., ... , Tq) be a q-tuple of mutually strongly commuting self-adjoint operators in a Hilbert space H. For A 2 0 and A> 0 the normed space 8..\;A(T) is defined by
S..\;A(T) = {! E D""(T) : II!IIT,..\;A = sup (IIT"JIIH A-lnl(lnl!t.\} < oo}. (2.7) nEN:
The Gevrey space S.\(T), brought about by T, is taken to be the union
SA(T) = u S.\;A(T) . (2.8) A>O
Clearly S.\(T) C Sp(T} if A < li· The topology for S..\(T) is the inductive limit topology generated by the normed spaces S.\;A(T), A > 0. Since the topology of s~(T) is regular, see [17, Theorem 1.11], we obtain the following characterization of sequential convergence in S.\(T). For a sequence (!~:heN C D""(T) we have
j,. -+ 0 ( k -+ oo) in S .\ (T) if and only if there exists A > 0 such that (2.9)
f~c E S,\;A(T), k E IN, and II/~:!IT.",A-+ 0 (k-+ oo).
Consistent with (2.7) and (2.8) we define Soo(T) D""(T). For a unitary operator U on H we have, cf. (1.43),
(2.10)
In particular we have for the Gevrey spaces S.~(P) and .S\(Q), brought about by the operator tuples P and Q, via the Fourier transformation,
(2.11)
cf. (1.48), (1.49). Moreover, for j = l, ... ,qand A E IRJ.oo we have
(2.12)
86
(2.13)
The results of the following lemma are known. We give an elementary proof; for ~ = oo, we refer to (1.50), (1.51).
Lemma 2.31 Let~ E ~00 • Then
9
(i) s~(Q) = s~(IQD = n s~(Q;) , i=1
" (ii) S~(P) = s~(IPI) = n S~(P;) . i=l
Proof. (i) We prove the inclusions
q
s~(Q) c n s~(Qi) c s~(IQD c s~(Q) . i=l
The first inclusion is obvious. Suppose J E S~(Q;), j = 1, ... , q. By using Newton's binomium and the Cauchy-Schwarz inequality we have for n = 2, ... , q and k E JIV0 ,
ll(xi + · · · + x!)"12 /II~(.R•) :5 E (~) ll(x~ + · · · + x!_t)i /IIL2(,Rq) llx!k-Zi /IIL2(.R•) · j=O 1
Then it follows by induction that
f E S~((Qi + ... + Q!)1'2), n = 2, ... , q.
Thus we have shown the second inclusion. The third inclusion follows from the inequality lx"l :5 lxjlkl , x E /Rq , k E IIVri. (ii) By applying Fourier transformation the assertion follows from (i). 0
The proof of the following lemma is similar to that of Lemma 2.10.
' Lemma 2.32 Let,\ E JR+ and j =I, ... ,q. Then
S~(IQI) = {/ E Lz(IR") : 3t>o exp(t lxl 1/~) f E Lz(/Rq)} ,
S~(Q;) {f E L2(JR") : 3t>o exp(t lx;J 11~)/ E L2(/Rq)}.
The proof of the following lemma is similar to that of Lemma 2.14; it is recalled that {\lln : nE JIVg} is the Hermite basis in L2 (JR"), as introduced in (1.53), (1.7).
Lemma 2.33 Let ,\ E JR+ and j = I, ... , q. Then
s~(IPI2 + IQI2) {! E L2(1R") : 3t>O (exp(t lnP 1~) (f, \II,.)L2(.R•)) E loo},
S~(P] + Qj) = {! E Lz(IR") : 3t>o (exp(t nY.\) (f, \II,.)L2(.R•J) E loo}.
Corollary 2.34 Let ~ E J1.+. Then q
'S>.([Pf,~ + IQf') = n S>.(PJ + QJ) . i=l
Proof. Supose f E S>.(Pj + QJ), j = 1, ... ,q. Due to the inequality
(m1 + m2 ) 11-' ::5 max(I,2(t/>.):...t) (mV" + ~1") , ml> ma E /No,
and due to Lemma. 2.33, it follows by induction that there exists t > 0 such that
(exp(t(nt + ... + n~c) 11") {!, Wn)L2(R•J) E loo, k = 2, ... ,q.
Hence, f E S>.([P[2 + IQI2 ) by Lemma. 2.33.
Obviously, S.~(IPI 2 + [Q[ 2) C S>.(Pl + Q1} , i = 1, ... , q.
We present the analogue of Lemma 2.5.
Lemma 2.35 Let f E S(JR'l). Then
(i) there exists Aq > 0 such that II/IIL2(R•) ::5 Aq 11(1 + [xl)q /IILoo(R•J ;
(ii) there exists Bq > 0 such that 11/IILoo(R•) ::5 B9 11(1 Ll)q /IIL2(R•J ;
mln(2lkl,llll q (2k ) (1 ) (iii) IIQkpl /lll2(R•) ::5 E ( n . n. J.nn in!) IIQ2/c-j /IIL2(R•) IIP2
I-j IIIL2(R•J ' l.ii=O n=l Jn
87
0
k,l E JN6.
Proof. (i) 11/IIL(R•) ::5 sup ([{1 + [xl)q /(x)[2 ) I (1 + [xl)-2q dx;
:z;ER• R•
(ii) (211')9n lf(x )I ::5 I i(ffi' f)(y)l dy = I i(ffi'(I- t.)q f) (y)l (1 + IYI2 )-q dy R• R•
(iii) since the self-a.djoint operators P; and Qi commute with each of the operators Pm and Qm if j :/; m, we derive by means of Leibniz' differentiation rule for k, l E JN6,
IIQ/cPI/IIL(R•) = ((P11 Q~kt Pf') .. · (P:•Q!Ic• P:•) /,/)L2(R•)
m.in(2lkl,l11) q ( 2k ) (1 ) E ilil( n .. n .n in!) (P21-j /, Q2k-j /)L2(R•) . l.il=O n=O Jn Jn
Now we obtain the wanted result by the Cauchy-Schwarz inequality. 0
We give some classical analytic characterizations of the spaces S'~( /Rq). For the corresponding characterizations of S(Bq) and S~(JR) we refer to Theorem 1.2:3 and Theorem 2.6.
88
Theorem 2.36 Let f E S(JR9). I. Let a E B;t. Then the following assertions are equivalent:
(i) 1 e s:un.q> ; (ii) there exist A, B; > 0 (j e IV0 ) such that llz'a' /III.,(Jiq) 5 BlliAI"I(Ikl!)'" , k, 1 e IV?, ;
(iii) there ezist A, B > 0 such that llz" fiiL:~(R•) 5 BAI"I(Ikl!)'" , k E IV3 ;
(iv) there ezist A, B > 0 such that lllxlk /IIL:~(Rt) 5 BA"(k!)'" , k E IVo ;
(v) there ezist A, B > 0 such that llx~fiiL:~(Rf) 5 BA"(k!)<> , j =I, ... , q, k E IV0 ;
(vi) there ezist A, B > 0 such that llx" /IIL .. (Rf) 5 BAI"I(Ikl!)<> , k e IVri ;
(vii) there exist A, B > 0 such that llxj/IIL .. (Rt) 5 BA"(k!)"' , j =I, ... , q, k E IV0 •
II. Let P e B;t. Then the following assertions are equivalent:
(i) 1 e se.,(JR.'I) ;
(ii) there ezist A;, B > 0 (j E IV0) such that llx"at /IIL:~(Rt) 5 AIA:IBI1I(Ili!)P , k, 1 e Jll/6 ;
(iii) there ezist A, B > 0 such that nat IIIL:~(R•) :::; ABI1I(Ili!)P ' I e INJ j
(iv) there ezist A, B > 0 such that 11~1/II14(Rf) 5 AB1(l!)P , lE Jll/0 ;
(v) there exist A, B > 0 such that ll&j/IIL:~(Rtl 5 AB1(1!)P, j = 1, ... , q, I e IV0 ;
(vi) there exist A, B > 0 such that 1181 /IIL .. (R') 5 ABI11(1li!)P, lE IN(, ;
(vii) there ezist A, B > 0 such that 118~/IIL .. (Rf) 5 AB1(l!)P , j = 1, ... , q, I e Jll/0 •
Ill. Let a, P E Rt with a+ P ;:::: 1. Then the following assertions are equivalent:
(ii) there exist A, B, C > 0 such that
(iii}there ezist A, B > 0 such that
(iv) there ezist A, B > 0 such that
89
( v) there ezist A, B > 0 such that
llxj/IIL,!R•J :5 BA"(k!)a , lla;/IIL,CR•l :5 AB1(l!)P, j = 1, ... , q, k,l E /No;
(vi) there exist A, B > 0 such that
(vii) there exist A, B > 0 such that
llxjfi!L .. (R•) :5 BA"(k!)<>, ll81fiiL..,(R') :5 AB1(l!)P, j = l, ... ,q, k,l E /No.
Proof. In each of the cases I, II and Ill, we follow the scheme (i) # (ii) # (iii) # (iv) #
(v), (i) =? (vi) =? (vii) =? (v). The assertions (iii), (iv) and (v) are mutually equivalent by Lemma. 2.31; the implications (ii) =? (iii), (i) =? (vi) =? (vii) need no explanation. The equivalence (i) # (ii) readily follows by means of Lemma. 2.35 (i) and (ii). The implication (iii) =? (ii) can be proved in the same manner as the implication (iii) =? (ii) in Theorem 2.6, using Lemma 2.35 (iii). Finally, the implication (vii) =? (v) follows from the estimates
IIQ1JIIl,(R•) = (Qj"/,/)L,(R') :511Qj"/IIL .. (R•) 11/IILI(R•), j = l, ... ,q, k E /No,
IIP]/IIl,(Rf) = (P]1f,f)L2(R') :5IIPl'fiiL .. (R•) 11/IIL,(R•), i = 1, .. · ,q, lE /No·
0
From Theorem 2.36 we obtain several results for the spac.es S~(IR'~). We start with the analogue of Theorem 2.7 (Kashpirovskii's intersection result).
Theorem 2.37 Let o:, {:J E JRt. Then
S';{IR'I) n S!,(JR9) S~(JRq) .
Proof. For a + {:J 2: 1 the identity immediately follows from Theorem 2.36. Next suppose 0 :5 a+{:J < 1. Let f E S';(lRq)nS!,(JRq). Then for fixed yE JRq-t the function x1 ~--+ f(x.,y) belongs to S';(JR)nS!(JR) by Theorem 2.:!6 I (vii), 11 (vii) and Theorem 2.6 I (iv ), II (iv). Hence/( Xt. y) = 0, x1 E JR, by Theorems 2. 7 and 2.2. Since y E JR9- 1
is arbitrary, f must be the null function. Thus we have shown that S;;>(JRq) n S{;,(JR'l) is the trivial space. Since, by Definition 2.30, S~(JR'l) C S:;;"(/R9 )n8{;,(1Rq), we are done. D
The action of the Fourier transformation F on S~(/Rq) swaps the indices o: and {:J; compare Theorem 2.8 and Corollary 2.9 (the proofs carry over).
Theorem 2.38 Let o:, {:J E JRt.w Then
F(S~(JRq)) = S8(JR9) ·
90
Corollary 2.39 Let a, fJ E .fl6 and let f : JR:l -+ C. Then the following assertions are equivalent:
(i) f E S~(Ul") ;
(ii) j E S!'(JR'I) and /Ff E Sif(R.9).
In Theorem 2.11 we proved the functional analytic characterization
S!(R.) = S,.(Q) n Sp(P), a,{J E .fl6.oo. By using the same techniques as in the proof of Theorem 2.11 and by means of Lemma 2.31 we obtain the following functional analytic characterization of S~(R.9) in terms of Gevrey spaces.
Theorem 2.40 Let a, fJ E JRt.w Then
q
S~{R.9) = s .. (Q) n Sp(P) = S,.(IQI) n S11(1PI) = n [S,(Qj) n Sp(Pi)]. i=l
As a consequence of Lemma 2.32 and Theorem 2.40, the functions in S:O(Ul") decrease exponentially at infinity; d. Theorem 2.12.
Theorem 2.41 Let a E JR.+, let p E {2,oo} and let f E S(JR'I). Then the following assertions are equivalent:
(i) f E S!'(R.9 ) ;
(ii) there exists t > 0 such that exp(t lxPI<>) f E L,(JR9) ;
(iii)there existst > 0 such thatexp(tlxJI11"')fE Lp(JRq), j = l, ... ,q.
The characterizations of S;:'(JR:l) established in Theorems 2.36 I and 2.41 lead, by Corollary 2.39, to analogous characterizations of S!,(JRq) and S~(/R9 ), a,{J E JR.+. Recalling the characterization of S(JR'I) by means of the Fourier transformation in Theorem 1.25, we obtain the following characterizations of the spaces se(/R9 ).
Theorem 2.42 Let f E L2(JR'I) and let p E {2,oo}. I. Let a E JR+. Then the following assertions are equivalent:
(ii) there exist A, B > 0 such that
(iii) there exist A, B > 0 such that
{iv) there exist A, B > 0 such that
llxjfiiL"(R9) :5 BA"(kl)" and x~Ff E L,(R'), j = 1, ... ,q, k,l E /No;
( v) there exists t > 0 such that
exp(tlxl1'")fE L,(IR11 ) and lxi'FfE L,(R'), lE /No;
(vi) there exists t > 0 such that
exp(t lxjl1'")f E L,(R') and x} Ff E L,(IR9 ), j = 1, ... , q, I E /No.
II. Let /1 E JR:+-. Then the following assertions are equivalent:
(i) f E S!,(R') ;
(ii) there exist A, B > 0 such that
x" f E L,( 1R11 ) lmd !lx11F f11Lp(R9) ~ ABI11(jlj!)11 , k, I E IN~ ;
(iii) there exist A, B > 0 such that
ixikf E L,(IR11 ) and lllxi11FfiiL"(.R9) ~ AB1(1!)11 , k,l E /No;
(iv) there exist A, B > 0 such that
xj/ E Lp(IR11 ) and llx}IFfi!Lp(R4) :5 AB1(1!)11, j = l, .. . ,q, k,l E /No;
( v) there exists t > 0 such that
lx lk f E Lp( IR'~) and exp( t lxl11il) F f E L,( IR11 ) , k E /No ;
(vi) there exists t > 0 such that
xjf E Lp(IR11 ) and exp(t lxiPI11 ) /Ff E Lp(R'), j = l, ... ,q, k E /No.
Ill. Let a, /1 E JR+. Then the following assertions are equimlcnt:
(i) f E S~(IR11 ) ;
(ii) there exist A, B > 0 such that
!lxk f11Lp(R9) ~ BAikl(lkl!)" and llx11F fi!Lp(Rq) :5 ABI'I(jlj!)13 , k, I E IN~ ;
(iii) there exist A, B > 0 such that
91
92
(iv) there ezist A, B > 0 such that
u~UIIL,.(.R9) ~ BA*(k!)" and u~~1P/IIz.,(.R9) ~ AB1(l!)P' j =I, ... ,q' k,l E /No i
( v) there emts t > 0 such that
(vi) there ezists t > 0 such that
As we have seen in Theorem 2.15, the functions in S~( IR) with o ?: ~ can be characterized in terms of their Hermite expansion coefficients:
s:(IR) = S2a(P2 + Q2) = {/ E L2(/R) : 3t>o (exp(tnt/(Zal) (/, '1/J .. )~(R}) E loo} .
Here {tPn : n E l/V0 } denotes the Hermite basis in Lz(IR) as introduced in (1.7). By a similar reasoning we obtain the analogue for S~(JR'I) with o?: !·
Theorem 2.43 Let o ?: ! . Then
q q
(i) s:(IRq) = n [Sa(Qi) n S,.(P;)] = n Sza(P} + Qj) = Sza(IPI2 + IQI2)
i=t i=t
{f E Lz(/Rq) 3t>O (exp(tn;'(Za)) (/, \II,.)~(Rq)) E loo , j = 1, ... , q}
= {f E Lz(/Rq) 3t>O (exp(tlnl1/(2"')) (/, \II,.)L2(JU)) E loo} .
(ii) For f E S!(/R9) we have
/(~) = L (/, \lln)~(.R9) \lln(~), X E /Rq , ne .NZ
where the series converges in S!(R).
Proof. (i) The first equality is taken from Theorem 2.40. By using the same techniques as in the proof of Theorem 2.15 we have S,.(Qj) n Sa(Pj) = S2,.(PJ + Qj), j = 1, ... ,q. The proof is completed by means of Lemma. 2.33 and Corollary 2.34. (iii) The proof is similar to the proof of Theorem 2.15 (ii). D
Finally, we present the analogue of Theorem 2.16 (the proof is the same).
Theorem 2.44 Let o E IR:J.w Then
s .. (IQI) n s .. (IPI) = Sz,.(IPI2 + IQI2) if and only if Q?: ! .
93
2.5 The spaces S~(JR'I) and expansions in spherical harmonics
In this section we want to carry over the results for the Schwartz space S(Ulq) established in Section 1.6, to the Gel'fand-Shilov spaces S~(ll9). We start with the characterization of the products of radially symmetric functions and homogeneous harmonic polynomials in sg(ll9). For the corresponding characterization of such products in S(Ul9) we refer to Theorem 1.37.
Theorem 2.45 Let a,{J E JRt.oo- Let g E Seven(R), let Y,. E ~ and define f: Ul9-+ C by
f(rw) = g(r) Ym(rw) ' r;::: 0 , wE sq-l .
Then I E sg(ll9) if and only if g E sg,even(JR.).
Proof. First we deal with the case of S:'Cffl9 ), 0 ~ a < oo. For a = 0, the equivalence follows from Theorems 2.36 I (iv) and 2.6 I (iii). Suppose 0 < a < oo. Since g E Seven(R), the following assertions are equivalent by Theorems 2.41 and 2.12:
- there exists t > 0 such that rm+q-1 exp(tr11")g E L2(JR.+);
- there exists r > 0 such that exp(rr11")g E L2(JR.+);
- 9 E s:,even(JR.) ·
Next we deal with the case of S!.,(ll9), 0 ~ P < oo. By Theorem 1.33 (Hecke-Bochner) we have (IF f) (rw) = ( -i)m (IHm+q/2-19) (r) Ym(rw), r > 0, wE sq-1 • Since g E Seven(lR.), the following assertions are equivalent by Corollary 2.39 and Theorems 2.41, 2.21 11:
there exists t > 0 such that rm+q-l exp(tr1113) IHm+q/Z-lU E L2(JR.+);
- there exists r > 0 such that exp(rr1113) 1Hm+qf2- 1g E L2(JR.+) ;
- 9 E S!o,even(JR.) ·
By means of the intersection result S;:'(ll9) n S!.,(JR.q) we obtain the wanted equivalence in the case of S~(JR.9 ), 0 s;::(JR.9) = S(JR'l) is covered by Theorem 1.37.
S~(JR.q), in Theorem 2.37, ~ a, /3 < oo. The case of
0
By taking m = 0 in Theorem 2.45, we obtain the following characterization of the radially symmetric functions in sg(Rq). For the corresponding characterization of the radially symmetric functions in S( JR.q) we refer to Theorem 1.36.
Theorem 2.46 Let a,/3 E JRt.oo· Let g E Seven(lR.) and define the radially symmetric function f : JR'l -+ C by
94
f(rw) = g(r), r ~ 0, wE 8 9-1
•
Then I E sg{JR9 ) if and only if g E sg,even{JR).
As y;e have seen in Theorem 1.38, the space S(/R!') can be characterized in terms of the expansion coefficients of its elements with respect to the basis { Un,m,j : n, m E JN0, j = 1, ... , N(q, m)}, introduced in Theorem 1.35, namely
S{lfl'l) = {/ E L2{/R9 ) : {{n +m)"{!, Un,mJ)~(.R•))n,m,j E loo , k E /No} .
Here is the analogue for the space S~(JR9) with a ~ ! {the proof is similar to the proofs of Theorems 2.15 and 2.43).
Theorem 2.47 Let a~!· Then
{i) S~{/R9 ) = {/ E L2{JR9) : 3t>o{exp{t{n + m)l/(2<>)) {!, Un,m,;)L2(.R•))n,mJ E loo} .
{ii) For f E S~(JR9) we have
oo N(q,m) f(x) = L L {!,un,m,j}L2 (.R•) Un,mJ(x), X E /R9 ,
n,m=O j=l
where the series converges in S~(JR9).
As we have seen in Theorem 1.39, a function f E S{/R9 ) admits the expansion
oo N(q,m)
f(rw) = E E J;,.,;(r) Ym,;(rw) , r ~ 0 , wE sq-l ,
m=O j=l
where the series converges in S{/R9 ) and where /m,j E Seven(IR) is given by
r'" /m,;{r) = j f(rw) Ym,;(w) du9-1(w), r > 0.
s•-1
For functions fin a Gel'fand-Shilov space we prove the following theorem.
Theorem 2.48 Let f E S(/R9 ).
I. Let a E JR+. Then the following assertions are equivalent:
oo N(q,m)
{ii) 31>0 L L 11 exp{trlfa) rm /m,;ll~.- 1 < oo . m=O i=l
11. Let /3 E JR+. Then the following assertions are equivalent:
oo N(q,m)
(ii) 31>0 L L 11 exp(trl//J) rm Dfm+q/2-tfm,;ll~.- 1 < 00 •
m=O j=l
(2.14)
{2.15)
Ill. Let a, {3 E JR+. Then the following assertions are equivalent:
(i) f E S!(JRq) ;
oo N(q,m)
(ii) 3t>O L L 11 exp(tr11"') rm /m,;llk._1 < oo and m=O i=1
oo N(q,m)
L L 11 exp(trl/13) rm Hm+q/2-tfmJIIk._1 < oo . m=O i=1
IV. Let a, {3 E JR:to and let f E S!(JRq). Then
fm,i E S~,even(/R) , mE /No , j = 1, ... , N(q, m) ,
and the series {2.14} converges in S!(JRq).
95
Proof. I. By Theorem 2.41 we have f E S';'(/Rq) if and only if there exists t > 0 such that exp(t lxl1'"') f E L2 (/Rq), whence we obtain the wanted equivalence. II. The equivalence follows from I and Theorems 2.38, 1.33 (Hecke-Bochner). Ill. The equivalence follows from I, II and Theorem 2.37. IV. The result follows from I, II, Ill and Theorems 2.12, 2.20 and 2.7. The convergence of the series (2.14) in S!(IRq) is obvious from I (ii), 11 (ii), Ill (ii). D
The remainder of this section deals with some preliminaries to the next section, where we study the Radon transformation on S!(IRq). In (1.71) we introduced the operator V as follows. Let f be a function on /Rq. Then V f is the function on Zq = lR X sq-1 defined by
(V!) (p,w) = f(pw), (p,w) E Zq.
We recall the characterization of the image V(S(/Rq)) by means of the operators Q@ I, P@ I and I@ !:lLB (see Theorem 1.43 and Definition 1.40): Let f be a function on ~. Then f E S( /Rq) if and only if
and V f satisfies the homogeneity conditions:
For lE /N0 , the function w t-t ((P1 @ I) V f) (0, w) is the restriction to sq-1 of a homogeneous polynomial of degree l on /Rq.
(2.16)
(2.17)
In the next theorem we present a characterization of the image V(S'';'(/Rq)) by means of the operators Q@ I, P@ I and I@ !:lLB. For the image V (S~ ( /Rq)) with {3 E JR+, we have not been able to establish such a characterization, cf. Theorem 2.50.
Theorem 2.49 Let a E JRt.oo and let f be a function on /Rq. Then f E S';' ( /Rq) if and only if
(2.18)
and V f satisfies the homogeneity conditions {2.17}.
96
Proof. First we observe that the case a = oo is covered by Theorem 1.43. Next, suppose f E S;:>(IR.'~) with a E JRt. Since S;:'(JR.'l) C $(Ul9 ), the function Vf satisfies (2.16) and (2.17). We show that V f E Sa(Q ® /). Due to Theorem 2.36 I (iv) there exist A, B > 0 such that for k E JN0 ,
j <J ll /(.JX<.~)I 2 dp)du9-
1(w)
s•-' .R
1 00
:52 j <J 1/(.JX<.~W dp + j IPA: f(.JX<.~W p9-1 dp) du9-
1 (w) s•-1 o 1
:5 20q 11/IIL.(.R•) + 2 lllxiA: /II~(.R•) :5 BAA:(k!)20 •
That is, V f E Sa(Q ® /). Conversely, suppose V f satisfies (2.18) and (2.17). We show that f E S;:>(Ul9). Since Sa(Q ® /) C D00 (Q ®/),the function Vf satisfies (2.16). Hence f E S(JR9 ). According to the definition of S01 (Q ®I) there exist A, B > 0 such that fork E /No,
00
j llxiA: f(xWdx = j <J lrk(Vf) (r,wWr9-1 dr)du9-
1(w) .Rq sq-1 o
1 00
:5 j <J lf(rwWdr+ j lrA:+9-1(Vf) (r,wWdr)du9-
1(w) sq-1 o 1
Since (k+q-1)! :5 2A:+9- 1 k!(q-1)!, it follows from Theorem 2.36 I (iv) that f E S;:>(Ul9 ). D
Theorem 2.50 Let a, f3 E JR:t.oo and let f E S~(Ul9 ). Then
and V f satisfies the homogeneity conditions {2.17}.
Proof. If f3 = oo, the result follows from Theorem 2.49. Suppose a = oo, f3 f. oo. Since S!(JR9 ) C S(Ul9), the function V f satisfies (2.16) and (2017)0 We show that V f E S13(P ® l)o For (p,w) E Z9 we have (8/op V f) (p,w) = w1(8d) (.JX<.~) +. o o + w9(89f) (.JX<.~). Hence for l E IN' it follows that ( 81 I 8p1 V f) (p, w) is the sum of q1 derivatives of the form wi ({)if) (.JX<.~) with j E IN;I, lil = lo So, due to Definition 2030 (ii) there exist A, B > 0 such that for lE /No,
j <J 1(81/8p1Vf) (p,wWdp) du9-1 (w)
sq-l .R
:511"09 sup (1 +p2) 1(81/olVJ) (p,wW :51r09 AB1(l!)213
o
(p,w)€Zq
'97
That is, V f E Sp(P ® J). Finally, suppose a -# oo, fl-# oo. Since S!(JR9) = S;-'(ll9 ) n se,(JR9 ), it follows from the preceding result and Theorem 2.49 that V f satisfies (2.17) and V I E S,(Q®l)nSp(P®l). We show that V I E S2o+2P(J ® ~LB)· For that purpose, we use the connection with the momentum operator M, see Theorem 1.42 (i),
(Mkf)(rw)=((I®.!li8 )VI)(r,w), r>O, wES9-1
;
we recall the definition of the momentum operator M from ( 1.64 ), namely
M = E (x;Oj - x;8;)2 •
1$i<j$q
By evaluation of the squares, it follows that M is the sum of 2q( q - 1) operators of the form x;10j1 X;2 8;-. with it.i2,it.h E {1,2, ... ,q}. Hence, fork E IN, it follows that M" is the sum of (2q(q- 1))" operators of the form n~=l Xi,.Ojm with im,im E {1, 2, ... , q} for m 1, 2, ... , 2k. By [17, Theorem 3.2] it follows that there exist A, B > 0 such that for k E !No,
IIMk /IIL2(R') $ BA"((2k)!)<>+P $ 8(22"'+2{3 A)k (k!)2a+2,8 .
Similarly, it follows that there exist A, B > 0 such that for k E /N0 ,
11(1- ~)9 M" IIIL2(R•) $ BA"(k!)2"'+2P .
By means of Theorem 1.42 (i) and Lemma 2.35 (ii) we estimate for k E /N0 ,
j (j I((I<8l~l8)VJ) (p,wWdp)do-9-1(w)
s,-1 R
I oo
$ 2 j <J I(M"f) (pw)l2 dp + j I( M" f) (pw)i2 pq-t dp)do-q- 1(w) gq-1 0 1
s 20q IIM" IIIL(Rq) + 2 IIMkiiil2(R•)
s 20q B; I I (I- ~)9 Mk IIIL2(RO) + 2 IIM" IIIL(R•) .
Combination of the latter three results yields V I E s2a+2,8(/@ LlLB ).
For the characterization of V(~(/R.9 )) with a> 1, we refer to Corollary 2.58.
0
By means of Theorems 2.49 and 2.50 we tackle the following factorization problem: For which functions g and h does the function (rw) ~---+ g(r) h(w) belong to S~(JR9)? We have not been able to completely solve this problem: The case I < ;1 < oo still requires further investigation. The solution of the above factorization problem in S'( IR9 ) can be found in Theorem 1.44. We recall that £.,. is the orthogonal projection of L2 (S9- 1 ) onto ~. see (1.78), and we observe that the spaces Qk(S~,even(ll)), k E Jfll, are described in Theo· rem 2.17 (ii).
Theorem 2.51 Let a, {l E JRt.00 , let g : [0, oo) -+ C such that g E X 9_ 11 let h : S9- 1 -+ C such that h E L2( sq-l) and define I : /R.9 -+ c by
98
f(rw) = g(r) h(w) , r;?;: 0 , wE 89-1
•
(i) If h = E~=0 E2m h with E2M h =I 0, then f E S~(JR'I) if and only if g E Q2M(S~even(.R)).
' . (ii) If h = E~=O E2m+l h with e2M+l h ¥ 0, then f E S~(/R9 ) if and only if g E Q2M+I(Sf,even(JR)).
(iii) If h cannot be represented by a finite expansion as in (i) or (ii},
then (1) f E S':(JR'I) if and only if
g E s:,_(.ll.) with g{1l(O) = 0, lE /No , and hE D00(~LB) ;
(2) f E S!(JR'I) with 0::;; {J::;; I, if and only if g = 0;
(3) f E S!(JR'I) with 1 < {J < oo, implies
g E S!,even(.ll.) with g!ll(O) = 0 , lE /No , and hE S2p(~LB) .
Proof. Suppose f E S!(JRn Since S!(JRq) c S(.ll'), it follows from Theorem 1.44 that g E Seven(.ll.) or g E Sodd(JR). Now, from Theorems 2.50 and 2.11 it follows that
g E S"(Q) n Sp(P) = S~(JR) and hE S2<>+2P(~LB). (2.19)
(i), (ii) The equivalences follow from (2.I9), Theorems 1.44 (i), (ii) and 2.45. (iii) (l) Suppose f E S';'(/RJI). Then it follows from (2.19) and Theorem 1.44 (iii) that g E S:,even(JR) with g<ll(O) = 0, lE /No, and hE D00(~LB)· Conversely, suppose g E s:,even(JR) with g(ll(O) = 0, l E /N0 , and h E D00(~LB)· Then f E S(JRq), by Theorem 1.44 (iii). According to Theorem 2.6 I (iii), there exist A, B > 0 such that fork E /N0 ,
00
j llxl" f(xW dx = j lh(wW duq- 1(w) j lr" g(r)l2 rq-l dr::;; BAk(k!)2".
Rq Sq-t o
Hence f E S':'(.ll'), by Theorem 2.36 I (iv). (iii) (2) Suppose f E S!(JRq) with 0 ::;; {J $ 1. Then it follows from (2.I9) and Theorem 1.44 (iii) that g E S!,even{JR) with g(ll(O) 0, lE /N0 . Since g can be extended to an analytic function on a strip IYI < 1/ B of the complex plane z = x + iy, by Theorem 2.3, we have g = 0 in that strip. This yields the wanted equivalence. (iii) (3) Suppose f E Sf(JR9) with 1 < {J < oo. Then it follows from (2.I9) and Theorem 1.44 (iii) that g E Sf,even(JR) with g(ll(O) = 0, lE /No. Let g E S£even(JR) and define J: .fl9-+ C by /(x) = g{lxl), x E .ll'. Then j E SC(lRq) by Theorem 2.46. Consider now the product J · j and write sC(JRq) = SQ"(lRq) n S!(.ll'). Obviously, j E S0 (JR'I) = G0 (1Rq) implies also J.J E S0 (.fl9). Starting from f E Sf(.ll') C S!(JR'I), j E Sf!,(JR9), it follows by means of Theorem 2.36 II (vii) that f · j E Sf!,(JRq). Thus we conclude that f. j E sC(JRq), that is, the function (rw) ~ g(r·) g(r) h(w) belongs to sg(.ll'). From (2.19) we then infer h E s2/3(~LB)· 0
The spherical harmonics {Ym,j: mE /N0 , j =I, ... , N(q, m)} constitute an orthonormal basis in L2(Sq-l) of eigenfunctions of the Laplace-Beltrami operator ~LB,
99
.6.LsYmJ = -m(m + q- 2) YmJ, mE .No, j = 1, ... ,N(q,m),
see (1.69). Therefore, the Gevrey space s~(.6.LB). brought about by .6.Ls, admits the following characterization:
Lemma 2.52 Let >. E JR+. Then
S~(.6.Ls) {hE Lz(S9-1
) : 3t>O (exp(tm21~) (h, YmJ)~(Sq-l))mJ E loo}.
The proof of Lemma 2.52 runs along the same lines as the proof of Lemma. 2.14. For the factorization problem in s:(JR.9) with a> 1, we present the following result.
Theorem 2.53 Let a > 1, let g E S(JR} and let h E D00 (.6.L8 ). Then the factorized function (rw) 1-+ g(r) h(w) belongs to S~(JR9 } if and only if there exists t > 0 such that
(exp(tnl/(2<>) + tm1/(2"')} (g, rmlL;:+q/2- 1 )x.,_1 (h, Ym,j )L2(SH j)n,m,j E loo .
Here JLr;:+q/2- 1 is the Laguerre function defined in (1.12).
Proof. The equivalence follows immediately from Theorem 2.47 (i) by observing that Un,mArw) = rm JLr;:+qf2- 1(r) YmJ(w), n, mE lNo, j = 1, ... 'N(q, m), 1' ;::: 0, wE sq-l' see Theorem 1.35, and by using the inequality
0
2.6 The Radon transformation on Sg(JRq)
In Section 1.7 we studied the Radon transformation on S(IR9 ). We recall the relationship between the Radon transformation 'R and the Fourier transformation IF, see Theorem 1.46,
(2.20)
The Fourier invariance of S(JR9 ) and the characterization of V(S(IR9 )) given in Theorem 1.43, led to the characterization of 'R(S(JR9)) in Theorem 1.47: Let g be an even function on Z9• Then g E 'R(S(/R9 )) if and only if
g E D00 (Q ® /) n D00(P ® /) n D00(/ ® flLs)
and g obeys the Helgason-Ludwig consistency conditions:
For l E .N0 , the function w 1-+ j t1 g( t, w) dt is the restriction n
to sq-l of a homogeneous polynomial of degree I on /Rq.
(2.21)
(2.22)
For the Gel'fand-Shilov spaces we have F(Sg(JRq)) = S(j(JR9), see Theorem 2.38, so that by (2.20) we obtain
(2.23)
100
In Theorem 2.49 we have been able to characterize the image V(Sp(_mq)) only for a= eo. Accordingly, we can characterize the image 'R.(S!(_mq)), only.
Theorem 2.54 Let (3 E JRt.oo and let g be an even function on Z9 • Then g E 'R.(S!(Dl9)) if and only if
g E D00 (Q ® /) n Sp(P ® /) n D00(/ ® ..:lLs)
and g obeys the Helgason-Ludwig consistency conditions (2.22).
Proof. According to Theorem 2.49 and (2.16), (2.17), we have
V(S;(_mq)) = V(S(Ul9)) n Sp(Q ®I) .
Hence, by means of (2.23) and (2.6) we deduce
'R.JS!,(Dl9)) =(IF*® I) (V(S(Dl9)) n Sp(Q ®I))= 'R.(S(Dl9)) n Sp(P ®I).
Now the theorem follows from (2.21 ), (2.22).
Theorem 2.55 Let a,(3 E JRt.oo and let g E 'R.(S~(Dl9 )). Then
g E Sa(Q ®I) n Sp(P ®I) n S2a+2P(/ ® ..:lLs)
and g obeys the Helgason-Ludwig consistency conditions (2.22}.
D
Proof. Since S~(Dl9 ) C S(Dl9 ), g obeys the Helgason-Ludwig consistency conditions (2.22) by Theorem 1.47. Further, by means of (2.23), Theorem 2.50 and (2.6), we deduce
'R.(S~(Dl9 )) =(IF*® I) V(S$(Dl9))
C (IF*® I) (Sp(Q ®I) n Sa(P ®I) n S2a+2P(/ ® ~Ls))
D
Helgason [19, Corollary 4.3] characterized the image 'R.('D(Dlq)), where 'D(Dl9) = S0 (Dl9), the space of infinitely differentiable functions on Dl9 with compact support. In our notation, Helgason's result reads as follows.
Theorem 2.56 Let g be an even function on Z9 • Then g E 'R.(S0 (_mq)) if and only if
and g obeys the Helgason-Ludwig consistency conditions (2.22}.
Corollary 2.57 Let (3 > I and let g be an even function on Z9 • Then g E 'R.(Sg(Dl9 )) if and only if
g E So(Q ® /) n Sp(P ® /) n D00(/ ® ~LB)
and g o'beys the Helgason-Ludwig consistency conditions (2.22).
101
Proof. Since SC(.fl'l) = S;'(El'l) n S!,(B"), the result follows from Theorems 2.54 and 2.56. 0
As a continuation of Theorems 2.49 and 2.50, we are now able to characterize the image V(,S:(.fl'l)) with a> 1.
Corollary 2.58 Let a :> I and let f be a function on lR". Then f E ,S:(IR") if and only if
V I E Sa(Q ® /) n So(P ® /) n D00 (1 ® ALB)
and V f satisfies the homogeneity conditions (2.17).
Proof. The result follows from (2.23), Corollary 2.57 and (2.6). 0
In the next theorem we characterize the range of the Radon transformation when applied to the radia.lly symmetric functions in S~(.fl'l). In this characterization the Erdelyi-Kober operators IVA, A E /R, defined in (1.23), appear. From Theorem 2.22 (ii) we recall that
(2.24)
For the corresponding characterization of the range of the Radon transformation when applied to the radially symmetricfunctions in S(/R9 ) we refer to Theorem 1.55.
Theorem 2.59 Let a,{3 E Bt.oo· (i) Let lo E s~.even(/R) and define the radially symmetric function I: /R" -t c by
/(rw) = /o(r) , r 2: 0, wE S"-1 •
Then f E S~(IR") and its Radon transform is given by
('Rf) (p,w) = (21f)(q-t)f2(1V(q-t)/do) (p)
P 2: o , w e s"-1 •
The Erdelyi-Kober transform IV{q-t)tdo belongs to S~·""""(JR).
(ii) Let go E S~,even(/R) and define the function g: Z, -t C by
g(p,w) = Uo(p) , (p,w) E Z9 •
Then g is the Radon transform of the function n.-t g given by
('R-1g) (rw) = (21ft<q-t)/2 (IV-{,-1)/29o) (r)
{
(21f)-(H)/2 (-r-1d/dr)<Hl/2 go(r) for q odd,
= 2(21f}-9/2
( -r-1d/dr)"'2 7 Uo(P) (p2- r2t 112 1Hlp I o1· q even,
r 2: 0 , wE S9 - 1 •
102
The Erdelyi-Kober transform lW-(9-l)/:IUo belongs to S!,_,..(JR.).
Proof. The theorem follows from (2.24) and Theorems 2.46 and 1.55. 0
At the end of this section we characterize the range of the Radon transformation when applied to functions I E s:(JR:l) of the form
f(rw) = f.,.(r) Ym(w) with /mE Qm(S!,even(IR.)), Ym E)%.
In this characterization the Weyl-Gegenba.uer transformation WQm,q/2_ 1 and its inverse WQ:~912_1 are involved. From (1.85), (1.86) we recall the definition
WQm,4/2-t = W_,.IWm+(q-tJ/2Q-m ,
WQ:~9t2_1 = QmlW_m-(q-l)/2Wm .
(2.25)
(2.26)
-Here the operators W.\, A E JR., are the Weyl operators defined in ( 1.34), while the operators lWA, A E JR., are the Erdelyi-Kober operators defined in (1.23). Since the Erdelyi-Kober operator IWm+(q-tJ/2 maps S~,even(JR.) onto itself, by {2.24), we have
(2.27)
In this connection we refer to Theorem 2.17 where we- have characterized the spaces Qm(se,even(JR.)) and pm(S~.even{JR.)). The operators WQm,9; 2_ 1 and WQ:~912_ 1 are integral operators with the Gegenbauer polynomial C~2-1 , defined in (1.88), in their kernels; see (1.89), (1.90). For the corresponding characterization of the range of the Radon transformation when applied to functions f E S(JR:l) of the form
/(rw} = fm(r) Ym(w) with /mE Qm(Seven(IR.)), Ym E Y! ,
we refer to Theorem 1.56.
Theorem 2.60 (i) Let fm E Qm(S!,even(JR.)), let Y,. E Y:, and define f: IR.9 ~ C by
f(rw) = /m(r) Y ... (w), r _;:: 0, wE S9-1
•
Then f E S!(JR.9 ) and its Radon transform is given by
('Rf) (p,w) = (211')(q-1)/2 (WQm,q/2-1 /m) (p) Ym(w)
00
= (211')(q-t)/2 j (r2 - p2)(q-a)f2 C';/.2- 1(p/r) fm(r)rdr Y ... (w), p
P :;::: o , w E sq-l •
The Weyl-Gegenbauer transform WQm,q/2-1 fm belongs to P"'(S'~,even(JR.)).
(ii) Let 9m E pm(s:,even(JR.)), let Ym E)% and define g: Z9 -+ C by
g(p,w) = 9m(P) Ym(w) , (p,w) E Z9 •
Then g is the Radon transform of the function 'R-1g given by
('R-1g) (rw) = (211't(q-l)/2 (W\i;.~9/2_1 gm) (r) Ym(w)
= ( -J21r)l-q !""' = (p2 _ r2)(q-3)/2 c'f!2-t(p/r) 9~-tl(p) dp Ym(w) , rq-2
r
The inverse Weyl-Gegenbauer transform wg;.~912_1 9m belongs to Qm(S~,even(JR)).
Proof. The theorem follows from (2.27) and Theorems 2.45 and 1.56.
103
0
104
105
CHAPTER 3
Fractional Calculus and Gegenbauer transformations
In C00 (JR+) we define the multiplication operators M,., a E JR., and the composition operators Tp, p E IR\{0}, by
(Maf) (x) = X 0 f(x) ' f E C00 (JR+) ' X> 0'
(Tpf) (x) = f(xP) , f E C00 (JR+) , X> 0 .
The operators Ma, a E JR., and Tp, p E IR\{0}, constitute one-parameter groups on C00 (JR+) with
The differentiation operator D defined by DJ= f' maps C00 (JR+) into itself, but is not a bijection. We mention the intertwining relations
3.1 The Weyl fractional operators
{3.1)
(3.2)
An appropriate space for establishing a fractional calculus based on Weyl operators is the space S ..... (JR+) which we define as follows.
Definition 3.1 The space S ..... (JR+) consists of all functions f E c=(JR+) for which
x" /(I) E L00 ([1, oo)) , k, l E /No .
In fact S ..... (JR+) consists of the C00 (JR+)-functions which, together with their derivatives, tend to zero more rapidly than any negative power of x as x -+ oo. There are no restrictions on the behaviour of the functions at x = 0. For instance, the function
f(x) = exp(l/x- x), x > 0,
belongs to S ..... (JR+). Along with the operators M 01 , a E JR., T,, p > 0, and D, which map S ..... (JR+) onto itself, we introduce the Weyl fractional integral operator W,. of order JL > 0 on S ..... (JR+), defined by
(W,.f) (x) = rt,) l f(t) (t- x)"-1 dt' f E S ..... (JR+)' X> 0' , > 0. (3.3)
106
It readily follows that W,. maps S-(B+) into C""(.fi+) with
DW,. = W,.D , p. > 0 .
Lemma 3.2 For p. > 0, the operator W,. maps S-(B+) into itself.
(3.4)
Proof. Let p. > 0 and let f E S-.(B+). The conclusion W,.J E S-(R+) follows from the estimate
00
jr(p) x"(D1W,.f) (x)l = lf(p.) x"(W,.D1f) (x)l = lx" j f(ll(t + x) t"'-1·dtl 0
00
$ j (t+x)" lfl'l(t+x)l (1 +(t+x)2"') t"'-1 /(1 +t2")dt
0
$ 27r sup le"(l + e~~) J!'l({)l , k, l € /No , X> 1 . p. ~>1
0
As usual in fractional calculus we want to extend the definition of the operator W11 to . non-positive values of p.. This can be done in a natural way: By an n times repeated integration by parts in the integral expression (3.3) for W11f we obtain
W11 = W,.+11(-D)", nE /No, p. > 0.
For fixed n E /N0 the right-hand side of the above equality makes sense for p. > -n. Therefore, we can extend the definition of the operator W,. to non-positive values of p. as follows:
(3.5}
Here rP.l denotes the smallest integer which is greater than or equal top.. Observe that W_, = (-D)" for nE IN. For this reason, the operator W_,., p. > 0, is called the Weyl fractional differential operator of order p.. The operators W,., p. E /R, constitute a group on s_(.fl+):
Theorem 3.3 For p., v E JR,
Proof. For p. = 0 or 11 = 0 the identity is trivial. Through<>ut the remainder of this proof it is understood that p., 11 > 0. For f E S-(JR+) and x > 0 we derive
. 1 ()() (W,.WJ) (x) = f(p.) ! (W.,f) (t) (t- x)"-1 dt
()() ()()
f(p./ f(v) ! <[ f(s) (s- w-1 ds) (t- x)"-
1 dt
107
00 •
= f(p./r(v) ! (! (s- w-l (t- x)~<-l dt) f(s) ds.
By substitution ofT = (t- x) / (s- x) in the inner integral, an Euler-type integral appears and we obtain
1 00
= f(p. + v) j f(s) (s- x)~<+v-l ds = (W,.+vf) (x) . .,
Through repeated integration by parts we find Wn(-D)n =I, nE /N0 • By use of this result and the relation (3.4) we derive
W_,.W~' = WfJ.<l-1'(-D)f~'l W~' = Wr,.1 _,.W,.(-D)f~<l = WrJ.tl(-D)f~<l =I,
and therefore,
W_,..W_v = Wr,.l-,..( -D)fpl Wrvl-v( -D)fvl = W-,.-vW,.+ .. Wfpl+fvl-,..-v( -D)fl'l+fvl
= W-,..-v Wr#l+fvl (-D)f,..l+fvl = W-,..-v .
Partitioning v Qr -p. into two appropriate parts we then infer that
{
w_,..wl'+(v-p) = w_,..wj.lwV-jl = Wv-u if P. ~ /) l
W_,..Wv = W-(p-v)-vWv = W_(,..-v)W-vWv = W.,_,.. if p. > v,
and finally,
W,..W-v = W~'Wfvl-v(-D)fvl = Wfvl- 11(-D)f"lW,.. = W_,W,.. = W,._,.
Since DM1 - M1D =I, it can be proved by induction (with respect to m) that
0
Here we prove the following generalization analogous to the second index law of Love [36, Sections 6,7].
Theorem 3.4 For p., v E JR.,
Proof. For p. 0 or v = 0 the identity is trivial. Throughout the remainder of this proof it is understood that p., /) > 0. For f E s_(JR+) and X> 0 we derive
108
1 00 00
= j (r_,.._, j f(t) (t- r)"-1 dt) (r- x)"-1 dr r(p) r(v) .
:z: r
1 00 t
= j f(t) <J (r- x)"-1 (t- r)"-1 r_,.._, dr) dt . r(p) r(v)
:z; :z;
By substitution ofT = (1/r -1/t) / (1/x -1/t) in the inner integral, an Euler-type integral appears and we obtain
, 00
= r(=~ v) j C" f(t) (t- x)"+"-1 dt = (M_,W,..+,M-,..1) (x). :z;
By rewriting the identity we also have
and by inversion we obtain
W_,M,..+,W-,.. = M,..W_,.._.,M,,
Thus all possible cases are covered.
By Leibniz's differentiation rule we have
DnM1 = M 1Dn + nnn- 1 , nE IN.
Here we prove the following generalization.
Theorem 3.5 For p E JR,
W,..Mt = Mt W,.. + 11 W,..+t .
0
Proof. For p = 0 the identity is trivial. Throughout the remainder of this proof it is understood that p > 0. For f E S .... (JR+) and x > 0 we derive
00 00
= rtp) ! f(t) (t- x)"-1
dt + r(/+ 1) ! f(t) (t- x)" dt
109
1 00
= r(p) ! (x + (t- z)) f(t) (t- z)"-1 dt = (W,.Mtf) (z) .
By use of this result a.nd Leibniz's differentiation rule we obtain
w_,.M. w_r,.1(Wr,.1-,.Mt) = w_r,.1(M. Wr,.l-11 + HPl - p) Wr,.1 -,.+~)
= (M. w_r,., - fpl w_r,.l+•> Wr,.,_,. + H11 l - p) w_,.+l =M. w_,. -~~ w_,.+l . 0
Next we introduce the Erdelyi-Kober operator IW,., I' E /R, on S_.(JR+), defined by
IW,. = 2-"T2 W,.Tt/2 , I' E lR .
For p > 0, this definition ca.n be elaborated by means of (3.3), viz.
(3.6)
The extension to non-positive values of p readily follows by means of (3.5), (3.2) and (3.1 ):
IW0 =I, IW_, = IWf,.l-,.(-2T2DT112)f"l = IW[,.]-,.(-M_ 1 D)f~'l , I'> 0. (3.8)
Observe that IW_n = ( -M-1 D)" for nE /N. For this reason, the operator IW_~'' p > 0, is called a fractional differential operator of order p. The preceding results for the operators WP, I' E /R, can be translated into results for the operators IWI', p E JR. Thus it follows from Lemma 3.2 that for p > 0, the operator IW,. maps S_,(JR+) into itself. The translation of Theorems 3.3, 3.4 and 3.5 is comprised in the following theorem.
Theorem 3.6 For p, v E /R,
(i) JWI'JW" = JWI'+v ,
By induction it can be proved that
D2"+t = (M-tD)" M2n+t(M-tD)"+l , nE /No.
The fractional version of this identity, given in the next theorem, is the key formula in our derivation of inversion formulas for Weyl-Gegenbauer transformations in Section 3.6.
Theorem 3. 'T For p E /R,
110
Proof. First we consider the case p. > -1. For I E s_(m+) and X> 0 we derive
(BV11+lM-z~>-tBV~>f) (x) = (BV~>+IM-2~>-tBV~>+t(BV-tf)) (x)
The inner integral can be evaluated by means of two formulas quoted from [18, 2.1.3 (10), 2.8 (6)], viz.
1
j t6-
1(1 w-b-t {1- zt)-"dt = B(b,c- b) 2F1(a,b;c;z), 0
Re c > Re b > 0 , lzl < 1 ,
F (a- 1 a·2a· z)- (1 + 1(1- z)tf2)t-2a 2 1 2' , , - 2 2 •
Indeed, by means of the substitution e = (r2 - x2) I (t2- x2 ) and the quoted formulas, we
have t
j (r2 - x2 )~' (t2 - r2 )~'r-2~'dr :z:
1
=i((t2-x2)/x)2~>+1 J e~>(t-e)~'(l-(l-t2/x2)e)-~>-1/2~ 0
= H(t2- x2)/x)2~'+I B(p. + 1, p. + 1) 2F1(p. + !, p. + 1; 2p. + 2; 1 - t2 /x2)
= 22~' B(p. + l, p. + 1) (t- x)2~'+t .
By combining the previous results we obtain <X>
(IW11+1M-2~>-tBV~>f) (x) = f(2
p.1 +
2) j (W-tf) (t) (t- x)2~'+t dt
. :z:
Thus we have proved the stated identity for p > -1. Next suppose p < 0, then -p.- 1 > -1. In the previous result we replace p by -p.- 1, then
W-2p-t = IW_~>M2~>+tiW-~>-t ,
and by inversion we obtain
111
This completes the proof of the theorem. CJ
Finally we introduce a subspace of S .... (m:+).
Definition 3.8 Let b E JR+. The space S .... b(JR+) consists of all functions J e Coo( m,+) for which
f(x) = 0, x >b.
Obviously S .... b(JR+), bE m.+, is a proper subspace of S .... (n+). Further we define
S .... oo(JR+) = S .... (JR+) .
The following statements can be verified easily.
Theorem 8.9 Let b, p E JR+ and let a, p. E JR. Then
(i) M~, WJ.l and 1WJ.l map S .... b(JR+) bijectively onto s_.(lR+) ;
(ii) Tp maps s ..... (JR+) bijectively onto s ..... IJp(m:+).
From (3.3) and (3.7) we infer that for p, > 0, the operators WJ.l and 1WJ.l on s ..... (JR+) are given by
1 b
(WI'f) (x) = f(p,) ! f(t) (t- x)"-1 dt' I E s_b(lR+) ' 0 <X< b' p, > 0 '(3.9)
21-~o~ b
(1WI'J) (x) = f(p,) ! f(t) (t2- x2)"-
1 tdt' I E s_b(lR+)' 0 <X< b' p, > 0.
(3.10)
3.2 The Riemann-Liouville fractional operators
An appropriate space for establishing a fractional calculus based on Riemann-Liouville operators is the space s_(JR+) which we define as follows.
Definition 8.10 The space S_(JR+) consists of all functions f E Coo( m,+) for which
l~ft; J(")(x) = 0 , k E DVo .
In fact we have
(3.11)
Along with the operators M", a E /R, Tp, p > 0, and D, which map S_(lR+) onto itself, we introduce the Riemann-Liouville fractional integral operator I ~t of order p, > 0 on S_(JR+ ), defined by
112
"' (/,.!) (x) = ft!J) f f(t) (x- w-l dt , f E S_(.fi.+) , X?> 0 , p > 0 .
' 0
(3.12)
Next we want to extend the definition of the operator I,. to non-positive values of p. By ann times repeated integration by parts in the integral expression (3.12) for I~'f we obtain
I11 = I .. +,.D" , n E /No , p > 0 .
Therefore, we can extend the definition of the operator / 11 to non-positive values of p as follows:
(3.13)
Observe that L.,. = D" for n E IN. For this reason, the operator L,., p > 0, is called the Riemann-Liouville fractional differential operator of order !J· We prove the following relation between the Riemann-Liouville operator I,. and the Weyl operator W,..
Theorem 3.11 For p E /R,
/" = M"-tT-1 W,.T-tMp+l .
Proof. If p > 0, the relation can be verified by a straightforward calculation based on the definitions (3.3) and (3.12) of Wl' and lw If p = 0, the relation holds, trivially. Next, it can be shown by induction that
M-n-tT-tD" = ( -D)"T-tM-n+t , nE IN.
By means of the previous results, Theorem 3.4 and (3.1), we derive for iJ > 0,
L,. = fr,.1 _,.Df~<l = Mr,.l-,.-tT-tWf,.l-~<M~<(M-f~<l-tT-IDf~<l)
= Mr,.l-~<-tT-t (Wr,.l-~<M,. W-f~<l) T_tM-f,.l+t
0
From Theorem 3.11 it follows that the operators / 11 , p E /R, map S_(.fi.+) bijectively onto itself. Furthermore, by means of Theorem 3.4 we can prove that the operators / 11 , p E /R, constitute a. group on S_(IR+). The following theorem contains the analogues of Theorems 3.3, 3.4 and 3.5.
Theorem 3.12 For p, v E /R,
(i) /p/v = /l'+v ,
Proof. (i) By means of Theorems 3.11 and 3.4, and (3.1), we derive
I~<.Iv = (Mp-tT-tWpT-tMp+I) (M..,_tT-tW ... T_tMv+I)
(ii) By means of Theorems 3.11 and 3.3 we derive
lpM-p-vlv = (Mp-tT-t WpT-tMp+I) M-p-v(M ... _1T_ 1 W ... T_ 1Mv+t)
(iii) By means of Theorems 3.11 and 3.5, and (3.1), we derive
/pMt- Mtlp = Mp-tT-tWpT-tMp+2- MpT-tWpT-tMp+t
Next we introduce the Erdelyi-Kober operator UP, J1. E /R, on 8-(IR+), defined by
U ~< = 2-~<T2/pTt/2 , J1. E lR .
For Jl. > 0, this definition can be elaborated by means of (3.12), viz.
113
0
(3.14)
The extension to non-positive values of Jl. readily follows by means of (3.13), (3.2) and (3.1):
(3.16)
Observe that U -n = (M_1Dt for n E /N. For this reason, the operator U -p• Jl. > 0, is called a fractional differential operator of order Jl.· By translation of Theorem 3.11 we obtain the following relation between the Erdelyi-Kober operators UP and M'p:
Theorem 3.13 For Jl. E /R,
Also the results in Theorem 3.12 for the operators JP, Jt E /R, can be translated into results for the operators U ~'' J1. E JR.
114
Theorem 3.14 For p., 11 E R,
(i) D,.Dv = D,.+v,
The next theorem provides the key formula in our derivation of inversion formulas for Riemann-Liouville-Gegenbauer transformations in Section 3.6; cf. Theorem 3.7.
Theorem 3.15 For p. ER,
Proof. By means of Theorems 3.11, 3.7 and 3.13, and (3.1), we derive
0
Finally we introduce a subspace of S .... (R+).
Definition 3.16 Let a E B,+. The space Sa .... (R+) consists of all functions f E C""(R+) for which
f(x)=O, O<x<a.
Obviously Sa .... (R+), a E JR+, is a proper subspace of S .... (m+). Observe that
(3.17)
Further we define
(3.18)
The following statements can be verified easily.
Theorem 3.17 Let a,p E B,+ and let a,p. E JR. Then
(i) MCJt, I,. and U,. map Sa .... (R+) bijectively onto Sa .... (JR+) ;
(ii) Tp maps Sa .... (R+) bijectively onto Sa,tP .... (R+) .
115
From (3.12) a.nd {3.15) we infer that for p. > 0, the operators 1,. and 1,. on S11 .... (JR+) are given by
1 ., (!,.!) (x) = r(p.) ! f(t) (x- w-l dt ' I E si1 .... (JR+) ' 0 <a< X , p. > 0' (3.19)
21-p .,
(1,./)(x)=r(p.) j f(t)(x2 -t2 )"'-ttdt, /ESo. .... (fR+), O<a<x, p.>O. 11
3.3 Rodrigues-type formulas for Gegenbauer functions
In Sections 3.4 and 3.5 we study four types of operator products, namely
W,.llV~, llV~W,., I,.Il~, //~I,..
(3.20)
These operator products ca.n be reduced to integral operators with Gegenbauer functions in their kernels. To prove this, we establish a formula for Gegenbauer functions which is an extension of the Rodrigues' formula for the Gegenbauer polynomials. We start with the introduction of some special functions. For p., v E /R the Legendre function Pf: is defined by means of the hypergeometric function 2Ft. cf. (18, 3.2(16)],
2-" (z + 1)"+~<12 z 1 Pf:(z)=r(l-p.) (z-l)"'/2 2Ft(-v,-v-p.;I-p.;z+ 1 , zEC.
The function Pf:(z) is known to be analytic in the complex z-plane cut along the real axis from -oo to 1. For z = x, x > 1, the above definition passes into
2-" (x+l)"+~</2 ( x-1) Pf:(x) = r(l- p.) (x- 1)"'/2 2F1 - v, -v- p.; 1- p.; X+ I ' X> 1 . (3.21)
The Legendre function on the cut -1 < x < 1 is denoted by IPt(x ). From the definition in [18, 3.4 (1 )] we have for p., v E IR,
lPt'(x) = Hei"'"''2 Pf:(x + iO) + e-'"'"''2 Pf:(x- iO)] (3.22)
2-v ( 1 + X )"+1'/2
= r(I _ p.) (I _ x)~</2 2Ft (- v, -v -1 <X< I .
The Legendre functions satisfy the relations (cf. [18, 3.3.1 {I), :JA (7)])
(3.23)
as can be verified by means of [18, 2.1.4 (23)], viz.
2Ft (a, b; c; z) = (1 - z)c-~1-b :~Ft ( c- a, c- b; c; z) , lzl < l .
For p., v E /R the Gegenbauer function C~ is defined in terms of the Legendre function by
116
(3.24)
This definition has been adopted from [18, 3.15.1 (3)] with all 'multiplicative constants omitted. Expressed in terms of the hypergeometric function 2F1, the Gegenbauer function is given by
2-v-p-1/2 Z - 1 c:(z)= f(p+l) (z+1)"2Fl(-v-p+l,-v;p+t;z+l), zEC,
which shows that C~(z) is analytic in the complex z-plane cut along the real axis from -oo to -1. For later convenience we introduce two notations for the Gegenba.uer function of real argument: c~ and g~ denote the retrictions of c~ to (1, 00) and (0, 1 ), respectively, so that
c:(z) = c:(x) = (x2 -1)1/4-P/2 p~~~'?t/2(x), X> 1 , (3.25)
I'!P(x) = C~'(x) = (1 - x2 ) 114-P/2 JP112-~' (x) 0 < x < I. ~., v v+p-1/2 '
Hence, by (3.23), the Gegenbauer functions satisfy the relations
c: = C~v-2p , !ie = g!:.v-2p •
(3.26)
(3.27)
For p > -~ with p f; 0 and n E /N0 , we define the Gegenbauer polynomial c:: by the. Rodrigues' formula, cf. (18, 3.15.1 (10)],
C~'(x) = (-2)" f(p + n) r(2p + n) (1- x2)1/2-,. (dfdx)" [(I- x~)n+p-1/2]. (3.28) " n! r(p) f(2p + 2n) '
Chebyshev polynomials and Legendre polynomials are special c~es of Gegenbauer polynomials. For p -+ 0 we obtain the Chebyshev polynomial of the first kind, T,., given by
C!(x) = lim ~ c:;(x) = ~ T,.(x) = ~ cos(n arccos x), nE IN. (3.29) ,. .... o p n n
For p = 1 we obtain the Chebyshev polynomial of the second kind, U,., given by
C~(x) = U,.(x) = sin((n + 1) arccos x) / sin(arccos x), nE /N0 • (3.30)
For p. = l we obtain the Legendre polynomial P,. defined by
C~l2(x) = P,.(x) = (2"n!)-1 (dfdx)" (x2 1)", nE /N0 • (3.31)
By [18, 3.15.1 ( 4)] we have the following relations between Gegenbauer functions and Gegenbauer polynomials: For p > - ~ with p. f; 0 and n E /N0 ,
C~(x) 2112-,. r(2p)n! C~'(x) (3.32) r(n + 2p.) r(p + 1/2) " '
C~(x) = /* T,.(x), (3.33)
I* C!(x) = --1
U,.(x) , n+
(3.34)
(3.35)
Furthermore, we observe that for I'> -~,
cg(x) = 21 12-~>f f(p + 1/2).
117
(3.36)
The following integral relations for Gegenbauer functions generalize the Rodrigues' formula for Gegenbauer polynomials.
Theorem 3.18 (i) For v E JR, Jl > 0, A > -~ and x > 1, we have
., rtp) 1 (t2 -1).\-l/2 c:(t) (x- w-~ dt = (x2 -1)>.+,.-t/2c:!:<x).
(ii) For v E /R, Jl > 0, A > -~ and 0 < x < 1, we have
1 1 -- j (1-t2)>.-tf:lg>.(t) (t-x)~'-1 dt (l-x2 )>..+,..-t/2 g;!;(x). r(p) " .,
Proof. (i) By substitution oft = x- (x- 1) s and by using the expansions
~ (a)k(b)k " d ~ (-d)t 1 2F1(a, b; c; z) = LJ ( ) k' z , (1 - z) = LJ - 11 - z ,
/c=O c k • 1=0 •
we derive
1
j s"'+l-t(l- s)>.+k-l/2 ds
0
x-1 )>.+v-1/2 --s x+l
The latter double series can he expressed in terms of the hypergeometric function 2 F1 •
Indeed, by means of the summation formula in Hansen [25, (7.4.15)], viz.
(c- b)n (c)n
118
we obtain
=E (-.\-v+pn(Jl)n (x-1)n t (-n)k(-v)k n=O (,\ + Jl + 2)n n! X+ 1 k=O (1- Jl- n)k k!
=E (-.\-v+~)~(Jl~V)n (x-1)n n=O (.\ + Jl + 2)n n. X+ 1
( X- 1) = 2F1 - A - v + 1 u - v· ,\ + u + !. --2'r ' r 2'x+ 1
Combination of the previous results yields the wanted integral formula. (ii) The proof of the second integral formula runs along the same lines. 0
To see that the integral relations in Theorem 3.18 generaliz~ the Rodrigues' formula for Gegenbauer polynomials, take Jl = v =nE IN in Theorem 3.18 and differentiate n times. Then the integral relations pass into
2-A-n+l/2 c;(x) = r(.\ + n + ~) (x2- 1)1/2-A (d/dxt [(x2- 1t+A-1/2],, X> 1 '
2-A-n+l/2 g;(x) = r(.\ + n + ~) (1- x2)1/2-A ( -dfdxt [(1- x2t+A-1/2] ' 0 < x < 1 '
which are equivalent to (3.32) and (3.28). Further Rodrigues-type formulas for Gegenbauer functions are presented in the following theorem.
Theorem 3.19 (i) For v E JR, nE /N0 , ,\ > n- ~ and x > 1, we have
c;+;:(x) = (x2 -l)n-A+l/2 (djdxt [(x2 -1)A-1/2c;(x)].
(ii) For v E /R, nE /No, ,\ > n -l and 0 < x < 1, we have
g;+;:(x) = (1-x2t-A+l/2(-d/dxt [(1-x2)A-1/2g;(x)].
Proof. For n = 0, the formulas are obvious. For n E IN, the stated formulas follow by setting Jl = n, ,\ := ,\- n, v := v + n in Theorem 3.18, and by differentiating the integral relations thus obtained, n times. 0
With the Rodrigues-type formulas for Gegenbauer functions at hand we are fully prepared to examine the four operator products mentioned in the beginning of this section.
119
3.4 The Weyi-Gegenbauer transformations
In this section we consider the Weyl operator products W"IW" and lW" W". We define the Weyl-Gegenbauer transformations in terms of these operator products. First we consider the product W"IW". As a. preliminary we present the following relation.
Lemma 3.20 For ..\, p. E JR,
W"IW" = W~<+2-'-tiW--'+1M2-'-t .
Proof. By means of Theorem 3.7 with p. =-..\we derive
W"IW" = W~<+2-'-t W_zA+tlW-' = W"+2~-t(IW-.\+tM2~-tiW-~) lW"
Theorem 3.21 Let..\, p. E lR with..\+ p. > 0. For f E S .... (JR+) we have
(W,.IW"J) (x) j (t2 x2 )"+~<-tg>:.!"-112(x/t) f(t)t 1-"dt, x > 0.
"'
0
Proof. First suppose ..\ > 0 and p. > 0. We start from the definitions (3.3) and (3.7) of W,. and IWx. By means of (3.36) with p. = ..\-1/2 and Theorem 3.18 (ii) with v 0 and ..\ replaced by A- 1/2, we derive for f E S .... (JR+) and x > 0,
00
= j (t2 - x2)"+"-1 fJ2!~'-112(xjt) f(t) t 1 -~' dt .
"' Next suppose ..\ > 0, p. :5 0 with ..\ + p. > 0. Choose n E IN such that p. + n > 0. Then by using the previous result and Theorem 3.19 (ii) with v = -p. n and A replaced by A+ p. + n- 1/2, we derive for f E S .... (JR+) and x > 0,
(W"IWxf) (x) = (W_ .. WI'+"IW"f) (x)
00
( -d/ dx)" J (t2 x2 )-'+~<+n-l g>:_t~!n-l/2(xjt) f(t) tl-p-n dt
"'
120
00
= j ( -dfd(zft))" [(1- x2/t2 )>.+p+n-1 g:!~!"-112(x/t)] l(t) t2A+P-1 dt "' .
00
= j (t2 - x2)A+P-1 g::p-112(xft) l(t) t 1-P dt .
"' Thus we have proved the wanted integral relation for A, p. E 81, subject to A > 0 and A + p. > 0. Finally, suppose A :5 0, p. > 0 with ..\ + p. > 0. Then by using Lemma 3.20, the previous result and (3.27) with v = -p. and p. replaced by A+ p.- 1/2, we derive for I E s_(JR+) and X > 0,
(WpM'>./) (x) = (W2>.+p-tM'-.HtM2>.-tf) (x)
00 J (t 2 - x2).Hp-l g~~c~~; (xft) l(t) tl-p dt
"' 00
= j (tz- x2)>.+p-I g~!p-1/2(xft) l(t) tl-p dt . 0
"'
We now come to the definition of the Weyi-Gegenba.uer transformation of the first kind, denoted by wg.,,..\, namely
(3.37)
Observe that wg.,,..\ maps S_(81+) bijectively onto itself. By means of Theorem 3.21 with p. -v and A replaced by v+..\+ 1/2, we obtain the following integral representation with a Gegenbauer function in its kernel.
Theorem 3.22 Let v,..\ E 81 with A> -1/2. For I E S_(81+) we have
00
(Wgv,:d) (x) = j (t2- x2 )>.-tf2 g;(xft) l(t)tdt, x > 0.
"'
Next we consider the product M'>. Ww As a preliminary we present the following relation; cf. Lemma. 3.20.
Lemma 3.23 For .A, p. E 81,
Proof. By means of Lemma 3.20 with p. replaced by -p. and,\ by -A, we derive
IW>.Wp = (W-plW->.)-1 = (W-~-~-2>.-tM'>.+tM-2>.-d- 1
0
121
Theorem 3.24 Let >., /J E R with >. + /J > 0. Then for f E S-(JR+) and x > 0 we have
00
(M'.\W1../) (z) = xl-1• 1 (t2- x2 )-'+~'-1 C;:!:':-tf2(t/x) f(t)dt .
.,
Proof. First suppose>.> -1 and /J > 1. Because of M'_1 = -M_1D = M_1 W_., we may write
whereupon we employ the definitions (3.7) and (3.3) of IJ.V.\+1 and W11_ 1• By means of (3.36) with /J >. + 1/2 and Theorem 3.18 (i) with v = 0, >. replaced by >. + 1/2 and /J by /J- 1, we derive for f E s_(m.+) and X> 0,
00
= x 1- 11 1 (t2 -x2)>.+,.-l c:::::-l12(tfx) f(t)dt. :c
Next suppose>.> -1, /J ~ 1 with>.+ IJ > 0. Choose nE 1N such that /J + n > I. Then by using the previous result, through integration by parts, and by means of Theorem 3.19 (i) with v = 1- /J nand>. replaced by>.+ /J + n 1/2, we derive for f E S_(R+) and X >0,
00
( -1)" x1_,_, 1 (t2- x2 ).\+~<+n-l c:::::::::-l'\t/x) /("l(t) dt :c
= x2.\+l'-l j (d/d(t/x))" ((t2/x2 -1).\+l'+n-l c;::::::::,•-l/2(t/x)) f(t)dt X
00
=x1 -~" 1 (t2 -x2 )>.+~"- 1 c;::::-112(tfx) f(t)dt. X
122
Thus we have proved the wanted integral relation for >., p. E /R, subject to ..\ > -I and ..\ + p. > 0. Finally, suppose ..\ :5 -1, p. > I with ..\ + p. > 0. Then by using Lemma 3.23, the previous result and (3.27) with v = I- p. and p. replaced by..\+ p. -I/2, we derive for f E S_(JR+) and·x > 0,
00
= x1-" J (t2
- x2 )~+1'-l c2ic;12(t/x) l(t) dt z
00
= x1-" j (t2 -x2 )A+~<-•c::::-112(t/x) l(t)dt. 0
X
We now come to the definition of the Weyl-Gegenbauer transformation of the second kind, denoted by we.,.~. namely
(3.38)
Observe that WC.,.~ maps S ... ( JR+) bijectively onto itself. By ·means of Theorem 3.24 with p. = 1 - v and ..\ replaced by v + ..\- 1/2, we obtain the following integral representation with a Gegenbauer function in its kernel.
Theorem 3.25 Let v, >. E lR with>.> -1/2. For I E S ... (JR+) we have
00
(WC.,.~!) (x) = x j (t2- x2 )~-t/2c;(t/x) f(t) dt, x > 0.
"'
Remark. The operators M01 , W,. and JJV~ map S-b(JR+) bijectively onto itself, see Theorem 3.9 (i). So both wg.,.~ and WC.,.~ map S_b(JR+) bijectively onto itself as well. Furthermore, in the integral representations for (Wg.,,AJ) (x) and (WC.,,.d) (x) of Theorems 3.22 and 3.25, valid for >. > -~, we may replace the upper limit oo by b whenever I E s ... b(JR+) and 0 < X < b.
3.5 The Riemann-Liouville-Gegenbauer transformations
In this section we consider the Riemann-Liouville operator products I,.DA and HAlw We define the Riemann-Liouville-Gegenhauer transformations in terms of these operator products. The derivations are omitted because they are fully analogous to the derivations in Section 3.4. First we consider the product I,.D>.. As a preliminary we present the following relation; cf. Lemma 3.20.
Lemma 3.26 For>., p. E /R,
1101>. = I,.+2>.-tD->.+IM2>.-t.
123
Theorem 3.27 Let.\, p E lR with,\+ p > 0. For f E S .... (JR+) we have
:t:
(Ip.lhf) (:~:) = I (x2 - t2)>.+,.-l c~t~<-l/2(x/t) f(t) tl-1' dt ' X > 0 . 0
The Riemann-Liouville-Gegenbauer transformation of the first kind, denoted by IC.,,>., is defined by
(3.39)
Observe that IC.,,>. maps S .... (JR+) bijectively onto itself. By means of Theorem 3.27 with p = -v and,\ replaced by v+ ,\ + 1/2, we obtain the following integral representation with a Gegenbauer function in its kernel.
Theorem 3.28 Let v,>. e lR with>.> -1/2. For J e S .... (JR+) we have
"' (ICv,>.J) (x):::: I (x2
- t 2 )>.-Jf•c;(xft) f(t) tdt, x > 0. 0
Next we consider the product ll >.I,.. As a preliminary we present the following relation; cf. Lemma 3.23.
Lemma 3.29 For >., p E JR,
Theorem 3.30 Let >.,p e IR with,\+ p > 0. For f E S,_(JR+) we have
:t:
(ll>.lp.f) (x) = x1-p. I (x2 t2)>.+,.-t Qt~:-112(tfx) f(t)dt, x > 0. 0
The Riemann-Liouville-Gegenbauer transformation of the second kind, denoted by IQ.,,>., is defined by
(3.40)
Observe that IQv,>. maps S,__(JR+) bijectively onto itself. By means of Theorem 3.30 with p = l - v and >. replaced by v + >.- 1/2, we obtain the following integral representation with a Gegenbauer function in its kernel.
Theorem 3.31 Let v,.\ e IR with>.> 1/2. For f e S .... (JR+) we have
"' (IQ,,>./) {x) x I (x2 t2)>.-tf•Q;(tfx) f(t)dt, x > 0.
0
124
Remark. The operators Ma, I,. and ll>. map S,.-(m+) bijectively onto itself, see Theorem 3.17 (i). So both IC.,,>. and IQ,,>. map s .. _(.Ul+) bijectively onto itself as well. Furthermore, in the integral representations for (IC.,,>.f) (x) and (IQ.,,>./) (x) of Theorems 3.28 and 3.31, valid for ,\ > -!, we may replace the lower limit 0 by a whenever f E B .. -( m+) and 0 <a< x.
3.6 The inverse Gegenbauer transformations
The Weyl-Gegenbauer transformations and the Riemann-Liouville-Gegenbauer transformations introduced in the previous sections, are henceforth called Gegenbauer transformations for short. In this section we establish several relations between these Gegenbauer transformations. Of special interest are the formulas for the inverse Gegenbauer transformations. These formulas will be used in the next section to solve certain integral equations. The following lemma is an easy consequence of the definitions (3.37)-(3.40) of the Gegenbauer transformations.
Lemma 3.32 For v, .\, p E JR,
(i) WpWQ.,,>.Mp = WQv-p,A+p,
Corresponding to (3.27) viz. c; = C~v-2>., Qt = Q~v-2>., we have the following counterparts for Gegenbauer transformations.
Theorem 3.33 For v, ,\ E /R,
{i) WQ.,,>. = WQ-v-2>.,>.,
(iii) IC.,,>. = IC-v-2>.,>. ,
Proof. We only prove the first identity. The proofs of the other identities run along the same lines. By means of {3.37) and Lemma 3.20 we deduce
0
The next theorem provides relations between the Weyi-Gegenbauer transformations and the Riemann-Liouville-Gegenbauer transformations.
125
Theorem 3.34 For v, .X E JR,
(i) wg.,,.\ = Mz.\T-tig.,,.\T-tM2>.+2 ,
Proof. (i) In the following derivation we use successively (3.37), Theorems 3.11, 3.13, 3.14 (ii), Lemma 3.26, Theorem 3.15, Lemma 3.29 and (3.40):
/
The proof of (ii) runs along the same lines. D
The next part of this section is devoted to the inverse Gegenbauer transformations. It is a straightforward consequence of the definitions (3.37)-(:lAO) of the Gegenbauer transformations that the inverse of a Gegenbauer transformation is again a Gegenbauer transformation.
Theorem 3.35 For v, A E JR,
(i) wg;,l = WCt-v,-t->. = WCv+2Mt,-t--' ;
(ii) WC;,l = Wgt-v,-1-.\ = Wgv+2-'+1,-l-.\ j
(iii) zc;,l = zgl-v,-1-J. = zgv+2A+l,-l-.\ ;
(iv) zg;,l = ICt-v,-1--' = ICv+2.\+l,-t-.\ .
Proof. We only prove the first identity. The proofs of the other identities run along the same lines. By means of the definitions (3.37), (3.38) of the Weyl-Gegenbauer transformations we have
126
The second equality in (i) follows from Theorem 3.33 (ii). 0
In theorems 3.22, 3.25, 3.28, 3.31 we have seen that the Gegenbauer transformations are integral operators if A > -!· Then it follows from Theorem 3.35 that the inverse Gegenbauer transformations are integral operators if A < -!· Next we show that the inverse Gegenbauer transformation can be expressed as the product of (fractional) differential operators and an integral operator.
Theorem 3.36 For v, A, p E /R,
(i) W\1;,1 = (M.,IW-pM-.. ) WCv+2.X-2p+t,-t-.x+P ,
(ii) WC;,i = W(1.,+2A-2p+t,-t-.x+p(Mt_.,IW_pMv-t) ,
(iii) IC;,i = (M.,lLpM_,)I\1.,+z-X-2P+t,-t-.x+p ,
(iv) I\1;,1 = ICv+2.X-2p+t,-t-.x+AMt-vl-pMv-d .
Proof. We only prove the first identity. The proofs of the other identities run along the same lines. From the definition (3.37) we have .
W\1v,.x = W_v1Wv+.X+t/2M-v = (W-.,IWv+A-p+t/ZM-v) M,,IWPM_,
W\1 ... .x-p(M.,IWpM_.,), p E lR.
Now the wanted result is obtained by inversion, using Theorem 3.35 (i). 0
In the next section we consider integral equations with Gegenbauer functions in their kernels. Various classical results for the solution of these integral equations will be explained in terms of the inversion formulas of Theorem 3.36. Alternative expressions for the solution can be obtained by means of the following inversion formulas for Gegenbauer transformations.
Theorem 3.37 For v, A E /R,
(i) wg;;;i = M-2.x-tWC .. ,.xW-z.x-t w-2.x-tWC,,.xM-n-t,
(ii) wc;,i = M-2A-tW\1,,.xW-2A-1 = W-2.X-tW\1v,.xM-2.X-I,
Proof. First we prove (i) and (iv) simultaneously. By means of (3.37), (3.38) and Lemma 3.23 we deduce
wg;;J = M,(IW-v-A-1/2Wv) = M_z.x-t(Mt-v1Wv+.X-t/2WI-v) w-2A-1
(3.41)
Similarly, by means of (3.39), (3.40) and Lemma. 3.26 we deduce
IQ;:J, = Uv-tlf-v->.H/2) Mv-1 = Lz>.-t(L,Jl.,+>.+J/2M_.,) M-2>.-t
= Lz>.-tiC.,,>.M-2>.-t .
By using Theorem 3.34, (3.42) and Theorem 3.11 we deduce
WQ;,l = M-2>.-zT-tiQ;JT-tM-2>.
= W_z>.-tWC.,,>.M-2>.-t.
Similarly, by using Theorem 3.34, (3.41) and Theorem 3.11 we deduce
IQ;:;i = M-z>.-zT-tWQ;;'JT-tM-2>.
= M-z>.-tiC.,,>.L2>.-t .
Thus we have proved (i) and (iv). By inversion we obtain (ii) and (iii).
127
(3.42)
0
We conclude this section with the presentation of Gegenbauer transform pairs. As we have seen in Theorems 3.35, 3.36 and 3.37, there are many formulas for the inverse Gegenbauer transformations. Here we make a particular choice. From Theorems 3.22, 3.25 and 3.37 (i), (ii) we obtain the following Weyl-Gegenbauer transform pairs.
Theorem 3.38 Let v,..\ E m with..\ > -!, and let b E mt.,. For f E S-.b(m+) we have
b
(WQ.,,>.f) (x) =I (t2- x2 )>.-ttzg;(xft) f(t)tdt, 0 < x < b,
"' b
(WQ;,~f) (x) = x-2" I (t2
- x2 )>.-tf2c;(tfx) (W-2>.-d) (t) dt , 0 < x < b;
"' and
b
(WC.,,,xf) (x) = x I (t 2- x2 )>.-t/2 c;(t/x) f(t) dt, 0 < x < b,
X
b
(WC;7,lf) (x) = x-z>.-t I (tz- x2 )>.-tfzg;(xft) (W-2,\-tf) (t)tdt, 0 < x <b. X
128
From Theorems 3.28, 3.31 and 3.37 (iii), (iv) we obtain the following Riemann-LiouvilleGegenbauer transform pairs.
Theorem 3.39 Let v, ..\ E IR with ..\ > -~, and let a E JRt. For f E S.,-(JR:+) we have
$
(ICv,>./) (x) = j (x2 t 2)>.-lf2c;(x/t) f(t)tdt, x >a,
" $
(IC;,\1) (x) = x-2>. j (x2- t2 )>.-t/zg;(t/x) U-2>.-d) (t) dt, x >a;
" and
"' (I(}v,>./) (x) = x j (x2
- t2)>.-t/2g:(t/x) f(t)dt, x >a,
" $
(IQ;,l!) (x) = x-2>.-t J (x2 - t 2)>.-l/2 c;(xft) U-2>.-1/) (t) t dt , X > a .
"
3. 7 Integral equations involving Gegenbauer functions
Based on the theory developed in the preceding sections of this chapter, we present a systematic method for solving Mellin-type convolution equations where the convolutor contains a Gegenbauer function. To be precise, we consider the integral equations
b
j (t2- x2 )>.-tl2 g;(x/t)t"+l g(t)dt = f(x), 0 < x < b; (3.43)
"' b
x"+I j (t2- x2)>.-t/2 c:(t/x) g(t) dt f(x) , 0 < x < b; (3.44)
"' X (
j (x2- tz)>.-t12 C;(x/t)t"+1 g(t)dt = f(x), x >a; (3.45)
" x-
x"+l j (x2- t2)>.-l/2 g;(t/x) g(t) dt = f(x) , x >a, (3.46)
" where v,p. E /R, ,\>-!,a E JR%, bE JRt,. In this section we discuss the integral equation (3.44) extensively. The other integral equations can be dealt with in the same manner. For convenience we assume/, g E S-o(JR+). By Theorem 3.25, the integral equation {3.44) can be written shortly as
M"WCv,>.9 =f.
129
Hence, the solution g is given by
u = wc;,lM-,.J . Now the various expressions for WC;,i, established in Theorems 3.35, 3.36, 3.37, provide integral/differential formulas for the solution g. The formulas of Theorem 3.35 (ii), viz. we;,~ = WOt-v,-1-A = WOvH,\+l,-1-,\ do not provide integral expressions for g, because -1- A < -!· From Theorem 3.36 (ii) with p E lR such that -1- A+ p > -1, and Theorem 3.22, we find
b
= 1 (y2-t2)p-A-3/2g~.j:~,\~2P+l(t/y)y2-vlW_p{y"-p-lJ(y)}dy' O<t<b. t
(3.47)
From Theorems 3.37 (ii) and 3.22 we obtain the following alternative expressions for g:
b
= t-2A-1 1 (y2- t2),\-1/2g;(t/y)yW_2A-dy-~< f(y)} dy, 0 < t < b' (3.48) t
and
b
= w-2,\-dj (y2- t2),\-lf2g;(t/y)y-2A-p. f(y)dy}, o < t <b. (3.49) I
The expressions for g presented follow as an immediate consequence of the inversion formulas for Gegenbauer transformations established in Section :J.6. We emphasize that the evaluation of the inversion formulas into integral expressions for g uses two basic ingredients, namely,
(1) Rodrigues-type formulas for Gegenbauer functions (Theorems :3.18, 3.19);
(2) intertwining relations between multiplication operators and fractional integral/ differential operators (Theorems 3. 7, 3.15).
For specific values of the parameters A, v, the Gegenbauer function kernel in the integral equation (3.44) reduces to a. Gegenba.uer polynomial or to one of the special cases, a Legendre polynomial or a. Chebyshev polynomial. Convolution integral equations with such polynomial kernels have been treated by Li 135], Buschman [4], [5], [6], and Higgins [30]. In these papers the solution of the integral equation seems to have been found by clever guessing, supported in 15] by a formal application of the Mellin transformation. The proposed solution is then verified by substitution into the original integral equation. This verification requires the (lengthy) computation of a special convolution integral. It is the aim of this section to show that the solutions of Li, Buschman, and Higgins, can be
130
found systematically as special cases of our solution (3.47) with specific values assigned to the parameters A, p., 11, p, b. In addition, we present alternative expressions for the solution of the integral equations considered, obtained by specialisation of (3.48) and (3.49). In the course of his work in aerodynamics, Li [35] was led to an integral equation with a Chebyshev polynomial of the first kind as its kernel. He showed that the integral equation
1
j (t2- x2
)-112 Tn(t/x) g(t)dt = f(x), 0 <X< I , (3.50)
has the solution
2 1
d -; J (y2- t2)-1f2Tn-I(t/y)yl-n dy (yn J(y)}dy' 0 < t < 1' t
g(t) (3.51)
provided f satisfies certain conditions. Here, in the case n = 0, T_1 is taken to beT~> and the integral equation (3.50) reduces to the classical Abel integral equation.
Now assume J,g E S_1(ffl+) and observe that Tn = ..;;J2c~, see (3.33). Then the Chebyshev transform pair (3.50), (3.5I) is precisely the special case of (3.44), (3.47) with A = 0, p. = 1, 11 = n, p = 1, b = 1. By assigning the same values to >., p,, 11, b in (3.48), (3.49), we obtain the following alternative expressions for g:
2 1 d . -; r 1 j (1/- t2
)-112 Tn(t/y)y dy (y f(y))dy, 0 < t < 1 ,
t
g(t)
2 d 1
. g(t) = -; dt {j (y2 -t2t 112 Tn(t/y)yf(y)dy} 1 0 < t < 1.
t
Inspired by Li's work, Buschman [4] considered an integral equation with a Legendre polynomial as its kernel. He showed that the integral equation
1
j Pn(t/x)g(t)dt=f(x), O<x<I, (3.52)
has the solution 1 (1 d)2 g{t) = j Pn-z(t/y)y2
-n Y dy (1t f(y))dy, 0 < t <I 1
l
(3.53)
provided f satisfies certain conditions. Now assume J,g E S_1(JR+) and observe that Pn C!12
, see (3.35). Then the Legendre transform pair (3.52), (3.53) is recognized as the special case of (3.44), (3.47) with A ~. p. = -1, 11 = n, p = 2, b = 1. By assigning the same values to >., p., 11, b in (3.48), (3.49), we obtain the following alternative expressions for g:
I ( d )2 g(t) = r 2 ! Pn(t/y)y dy (yf(y))dy, 0 < t < 1,
( d )2 1
g(t) dt {j Pn(t/y) J(y)dy} 1 0 < t < 1 . l
131
Next, Buschman [5] considered an integral equation involving the Gegenbauer polynomial C!/2 with k, n E /No, 0 ~ k < n. For convenience we rewrite C!/2 in terms of our notation C!f2
, cf. (3.32). Then Buschman 's integral equation takes the form
1 f (t2 - x2)kf2-l/'2 C!f2(t/x) g(t) dt = l(x) , 0 <X< 1 , (3.54)
with the solution
g(t) == i (y2- t2)k/2-t/2c~'!k-t(t/y)y2-n (-!.. dd )k+t (yn l(y))dy' 0 < t < 1' t y y
(3.55)
provided I satisfies certain conditions. Now assume l,g E S_1 (JR.+). Then the Gegenbauer transform pair (3.54), (3.55) is precisely the special case of (3.44), (3.47) with >. = k/2, p == -1, v == n, p = k + 1, b = l. The corresponding specialisation of (3.48), (3.49) yields the following alternative expressions for g:
Buschman's transform pair (3.54), (3.55) covers Li's transform pair (3.50), {3.51) (take k == 0) and Buschman's transform pair (3.52), (3.53) (take k = 1). Higgins [30] studied an integral equation involving the Gegenbauer polynomial C! with m E IN, >. > -!· Again for convenience we rewrite C! in terms of our notation C~, d. (3.32). Then Higgins' integral equation takes the form
1
j (t 2- x2)A-tf2 C~(t/x) g(t)dt = l(x), 0 < x < 1, (3.56)
with the solution 1
g(t) == j (y2 - t2)"-1' 2c:(t/y)y2-miW_"_"_t{ym l(y)} dy, 0 < t < 1, (3.57) t
where n,m E IN, n <m, p ==(m- n -1)/2, >.>-!,provided I satisfies certain conditions. Now assume l,g E S_1(JR+). Then the Gegenbauer transform pair (:3.56), (3.57) is recognized as the special case of (3.44), (3.47) with>. == >., p == -1, v = m, p = >.+(m -n+ 1 )/2, b == l. The corresponding specialisation of (3.48), (3.49) yields the following alternative expressions for g:
132
1
g(t)=t-2>.-1 j (y2 -t2 )>.-tf2 C~(tfy)yW_2.\-dYf(y)}dy,, O<t< 1, t
1
g(t) = w-2.\-dj <vz- t2).\-1/2c~(t/y)y•-2.\ f(y)dy}, o < t < 1. t
Higgins' transform pair (3.56), (3.57) covers Buschman's transform pair (3.54}, (3.55) (take ..X= k/2, n =m- k- 1, so that I' :::: k/2). Finally, Buschman [6] considered an integral equation with the Legendre function p;-~~o as its kernel. He showed that the integral equation
1
j (t2 - x2)(P.- 1 )tzp;-~~o(tfx) g(t)dt = f(x), 0 < x < l , (3.58) X
has the solution 1
g(t) J (y2 t2)(n-p.-l)/2 JP;::;:+P.(t/y)y"-11 ( -; :v) n (y" J(y))dy' 0 < t < 1'
(3.59)
where n E IN, n > p > 0, provided f satisfies certain conditions. Now assume f,g E S_1(JR+) and observe that p;-~~o(z) = (z2 - l)l~~o-t)/2 c=::::!Z1 (z), cf. (3.24). Then the Legendre transform pair (3.56), (3.57) is precisely the special case of (3.44), (3.47) with ..X := p- ~~ Jl := -fl, 11 := 11- Jl + 1, p = n, b = 1. By assigning the same values to ..X, p, v, b in (3.48}, (3.49), we obtain the following alternative expressions for g:
1
g(t) = C2~' j (y2 - t2 )1~~o- 1 J/2 IP;-~'(tfy) y~' W-2~~o{Y~' f(y)} dy , 0 < t < l , t
1
g(t) = W_z~~o!] (y2 - t2 )1~~o-t)/zJP;-~~o(t/y) f(y)dy}, 0 < t < 1 . I
From (3.47) it follows that the expression (3.59) for g remains valid for non-integral n > fl, provided that ( -y-1 dfdy)" is replaced by ]W_,.. Thus, Buschman's transform pair (3.58), (3.59) covevs Higgins' transform pair (3.56), (3.57) (take p := >. + ~' 11 := m+>.-~' n :=..X+ (m n + 1)/2, f(x) := x-.\+1/2 f(x)). We briefly discuss two further papers on the solution of convolution integral equations. Erdelyi [20] considered an integral equation involving the Legendre function P;>., which he solved by techniques based on Riemann-Liouville fractional calculus. Erdelyi's integral equation is equivalent to our equation (3.45) and can be written shortly as
ICv->.,H112M->.-tY = J .
His solution can he identified with one of the many expressions for M>.+IIC;~>.,>.+t!d• obtainable from Theorems 3.35, 3.36, 3.37. Alternative representations for the solution g, analogous to (3.48) and (3.49), can be derived as well.
133
Sneddon [47} treated the general Mellin-type convolution equation which he solved by Mellin transform techniques. He recovered the previous solutions of Li, Buscbman, Riggins, and Erdelyi, as special cases. However, his procedure is not very transparent and requires a thorough knowledge of Mellin transforms. For a detailed review of the literature on convolution integral equations with special function kernels, we refer to Srivastava. and Buschman [48]. More recently, Deans [7], [8] utilized the Radon transformation to determine a. Gegenba.uer transform pair with the Gegenba.uer polynomial C!,{2- 1, m, q E JN, q ;:: 2, as its kernel. For convenience we rewrite C!,(2- 1 in terms of our notation C~{l- 1 , cf. (3.32). Then Deans' Gegenbauer transform pair takes the form
00
f(~) = j (t2 - x2)(q-J)f2 C':{.2- 1(x/t) tg(t)dt, x > 0, (3.60} .,
00
g(t) = t2-q J (y2 (3.61) t
where m,q E IN, q 2 2, provided f satisfies certain conditions. Now assume J,g E S_(Dl+). Then the Gegenba.uer transform pair (3.60), (3.61) is identical to the first transform pair in Theorem 3.38 with A q/2- 1, v = m, b oo. From Theorem 3.37 (i) with A = qf2- 1, v = m, we obtain the following alternative expression for g:
( d )q-l 00
g(t) = - dt {t / (y2 -t2)(q-3J/2c'ft.2-l(y/t)yl-q J(y}dy}, t > o.
The Gegenbauer transform pair (3.60), (3.61) has been used in Sectionl.7, formulas (1.89), (1.90), and in Section 2.6. In these sections we studied the Ra.don transformation 'R acting on functions f E S(Dl9 ) or E S~(Dl9 ), of the form
f(rw) fm(r) Ym(w), r;:: 0, wE 89-1
,
where r-m f E Seven(Dl) or E s~,even(JR) and Ym E Y~- It was shown that
('Rf) (p,w) {21f)(q-l)/2{WQm,q/2-tfm) (p) Ym(w)
= (21f)(q-l)/2 7 (r2 -l)(q-3)/2 c':{.2-1(pfr) r f,..(1·) (b· Y,,.(w) ' p
P ~ o , w e sq-l •
Conversely, let the function g Z9 -+ C be defined by
g(p,w) Ym(P) Ym(w) , (p,w) E Z9 ,
where 9m E P"'(Seven(Dl)) orE pm(S~,even(Dl)) and Y,, E Y,~,. Theu we have the inverse Ra.don transform
('R-1g) (rw) = (21ft(q-l)/2 (WQ;.\;2_ 1gm) (r) Y,,.(w)
( -.J21r)' -q loo = (p2 - r2)(q-3)/2 c':{.2-l(p/1') o!,~-ll(p) dp Y,,.(w) , rq-2
T
,. ;:= o , w e sq-l . These results ha.ve been stated in Theorems 1.56 and 2.60.
134
135
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138
GLOSSARY OF SYMBOLS
Special functions
Gegenbauer function, 116 restriction of ce to (1, 00 ), 116 restriction of ce to (0, 1 ), 116 Gegenbauer polynomial, 116 Hypergeometric function, 117 Hermite polynomial, 12 Hermite function of one variable, 12 Hermite function of q variables, 34 Bessel function of the first kind, 17 generalized Laguerre polynomial, 15 Laguerre function, 15 Fourier-Laguerre function, 24 Legendre polynomial, 116 Legendre function, 115 Legendre function on the cut, 115 Chebyshev polynomial of the first kind, 116 Chebyshev polynomial of the second kind, 116
Basic symbols
12 loo L2(X) Loo(X) JR+
IRt IR!, IRJ.oo /Rq
x·y lxl xk
IY aj INci In I sq-1 uq-J
nq Zq XII Y! N(q,m)
space of square summable sequences space of bounded sequences space of square integrable functions on X space of bounded functions on X set of positive real numbers JR+ u {0}, 61 JR+ U {oo}, 61 JR+ U {O,oo}, 61 q-dimensional Euclidean space Euclidean inner product in ~, 34 (x·x) 112,34
kl k2 kq 85 x 1 x 2 • ••• • Xq ,
a!~a~2 ••••• a!•, 85 a;axj, 85 set of q-tuples of nonnegative integers n1 + n2 + ... + nq, 34 unit sphere in /Rq, 34 measure on sq-J' 34 qq-l(sq-l), total surface area of sq-l, 35 JR X Sq-J, unit cylinder in JRq+l, 44 L2(JR+; x" dx), 15 space of spherical harmonics of degree m on /Rq, 36 number of linearly independent elements in )'!, 36
Kq m
Ym,j YmJ Un,mJ(rw) ®
r"'1 0
Operators T* D(T) D""(T) S>.(T) T TA:
ITI D""(T) SA(T) Q,Qj,Q P,P3,P :F, :Fi, IF, IF; H ... H-t/2 1-lv 1-l x·\7 M ~
~LB V 'R,
Cm Msgn
M~
T" D w~-~ IWP /I-' 111-' wgv,>. wc .... >-
zc ... ,A
Igv,>.
reproducing kernel of Y!., 38 spherical harmonic of degree m on JR9, 39 harmonic extension of YmJ• 36 JL:'+q/2- 1 (r) YmJ(rw), 39 tensor product, 44 smallest integer greater than or equal top, 106 end of proof
adjoint of operator T domain of operator T 0""-doma.in of operator T, 10 Gevrey space. brought about by operator T, 67 q-tuple of operators (T1 , ••• , T11 ), 30 rt'r? ... r;q. 30 (T{ + Ti + ... + Ti)t/2, 30 coo -domain of T, 30 Gevrey space brought about by T, 85 multiplication operators, 10, 30, 31 differentiation operators, 10, 30, 31 Fourier transformations, 9, 10, 30, 31 Ha.nkel transformation, 17 Fourier-cosine transformation, 17 Hankel-Clifford transformation, 24 Hilbert transformation, 52 differential operator, 39 momentum operator, 39 Laplacian, 31 Laplace-Beltrami operator, 41 connection operator, 46 Radon transformation, 50 orthogona.l projection of L2(Sq-t) onto Y~, 49 multiplication by the function sgn, 52 multiplication operator, 105 composition operator, 105 differentiation operator, 105 Weyl operator, 105, 106 Erdelyi-Kober operator, 109 Riemann-Liouville operator, 111, 112 Erdelyi-Kober operator, 113 Weyl-Gegenbauer transformation of the first kind, 120 Weyi-Gegenbauer transformation of the second kind, 122 Riemann-Liouville-Gegenbauer transformation of the first kind, 123 Riemann-Liouville-Gegenbauer transformation of the second kind, 123
139
140
FUnction spaces C<»(X) C8"(X) S(JR) Sev~{JR) sodd(IR) S(JR+) s_(JR+) s_b(JR+) S,_(JR+) Sa,_(JR+) s ..... (JR+) S(JR:I) S(Zq) Seven(Zq) sodd(Z9 )
S~(IR) s~ even(/R)
rl s<x,odd(IR) G~(JR+) S~(JR9)
space of infinitely differentiable functions on X subspace of C""(X) of functions with compact support Schwa.rtz space on IR, 9 subspace of even functions in S(IR), 14 subspa.ce of odd functions in S(JR), 14 Schwa.rtz space on JR+, 23 subspace of C""(JR+), 105 subspace of s_(JR+), 111 subspa.ce of C<»(JR+), 111 subspace of S ..... (JR+), 114 s_(JR+) n s_(JR+), 26 Schwa.rtz space on JR:I, 29 Schwartz space on Zq, 45 subspace of even functions in S(Zq), 46 subspace of odd functions in S(Z9 ), 46 Gel'fand-Shilov space on IR, 61 subspace of even functions in S~(IR), 74 subspace of odd functions in S~(IR), 74 Gel'fa.nd-Shilov space on JR+, 78 Gel'fand-Shilov space on JR9, 84, 85
INDEX C""-domain, 10 Cauchy problem, 55, 56 Chebyshev polynomial, 116, 130 Composition operator, 105
Differentiation operator, 10, 30, 31, 105
Erdelyi-Koher operator, 20, 57, 75, 101,
109, 113 Expansion in spherical harmonics, 41, 93
Factorization problem, 49, 97
Fourier transformation, 9, 10, 30, 31 Fourier-cosine transformation, 17 Fourier-Laguerre function, 24, 25, 79
Fractional calculus, 105 Functions of rapid decrease, 9 Funk-Hecke theorem, 37
Gegenbauer function, 115 Gegenbauer polynomial, 37, 116
Gegenbauer transformation, 124
Gel'fand-Shilov space, 61, 84
Gevrey space, 67, 85
Hankel transformation, 17, 74
Hankel-Clifford transformation, 24, 78 Harmonic polynomial, 36 Hecke-Bochner theorem, 38 Hermite function, 12, 34, 70, 92 Hermite polynomial, 12
Hilbert transformation, 52
Homogeneous polynomial, 36
Huygens' principle, 57 Hyperplane, 50
Integral equation, 128 Intertwining relation, 105, 129
141
Inverse Gegenbauer transformation, 124
Ka.shpirovskii's intersection result, 65, 89
La.guerre function, 15, 74
Laguerre polynomial (generalized), 15
La.placian, 31 La.place-Beltrami operator, 41, 46 Legendre function, 115 Legendre function on the cut, 115 Legendre polynomial, 116, 130
Mellin-type convolution equation, 128
Momentum operator, 39, 46 Multi-index notation, 29, 85
Multiplication operator, 10, 30, 31, 105
Projection theorem, 51
Ra.dially symmetric function, 41, 5~?' 93, 101 Radon inversion formula, 52, 53, 54, 55, 58,
59 Radon transformation, 50
Reproducing kernel, 38 Riemann-Liouville operator, 111 Riemann-Liouville-Gegenbauer
transformation, 122, 123
Rodrigues' formula, 115, ll6, 117, ll8
Schwartz space, 9, 29 Spherical harmonic, :34, 36
Tensor product, 44
Weyl operator, 25, 79, 105 Weyi-Gegenbauer transformation, 58, 59,
102, 119, 120, 122
Zonal spherical harmonic, 37
142
SAMENVATTING
In dit proefschrift wordt de werking van een aantal klassieke integraaltransformaties op de functieruimte S~(JR'l), a,{3 E JRt.oo, q E IN, bestudeerd. De notatie S~(JR'l) staat voor Gel'fand-Shilovruimte met als bijzonder geval de Schwartzruimte s:(JR'l) = S(JR'l) der snel afnemende functies. Genoemde functieruimten kunnen gekarakteriseerd worden met behulp van de zelfgeadjungeerde operatoren Qi (vermenigvuldiging met x;) en Pi = iofoxi, j = 1, ... , q, in L2 (JR'l). Uit deze karakteriseringen volgen de bekende fraaie eigenschappen
waarin IF de Fouriertransformatie op L2(JR'l) is. Tevens is de ruimte S~(JR'l) te karakteriseren met behulp van de Fouriertransformatie. In geval q = 1 is de ruimte S~,even(Dl), i.e. de ruimte der even functies in S~(Dl), te beschrijven in termen van de Hankeltransformatie lfv:
llv(S~,even(R.)) = Sp,even(Hl), a,{3 E JRt.oo, V;?:-~.
Met behulp van deze ka.rakteriseringen en het bekende verband tussen de Fouriertransformatie en de Hankeltransforma.tie (stelling van Hecke-Bochner) wordt de volgende karakte- · risering van de radiaalsymmetrischefuncties in S~(.lllq) afgeleid: Een functie f(x) = g(jxl) behoort tot S~(JR'l) dan en slechts dan als g E S~,even(Dl). Door ontwikkeling naar spherische harmonischen wordt de overeenkomstige karakterisering van de niet-radiaalsymmetrische functies in S~(JR'l) gevonden. Tenslotte wordt de werking van de Radontransfor~atie op S'[;(.Ulq) onderzocht. De gekozen aanpak is gebaseerd op een operator-calculus. Hierbij spelen zowel het verband tussen de Radontransformatie en de Fouriertransformatie als het verband tussen de Radontransformatie en de Weyl-Gegenbauertra.nsformaties een belangrijke rol. Laatstgenoemde tra.nsformaties behoren tot de klasse der Gegenba.uertransforma.ties, waarvan /de theorie ontwikkeld wordt in Hoofdstuk 3. De Gegenba.uertransforma.ties zijn composities van fractionele integraal- en differentiaaloperatoren van het type Weyl, Riema.nn-Liouville of Erdelyi-Kober. Deze transformaties vormen het passende gereedschap voor de systematische oplossing van zekere integraalvergelijkingen van het Mellin-convolutietype.
CURRICULUM VITAE
27 februari 1964 g~boren te Sint-Oedenrode
4 juni 1982 eindexamen Gymnasium {1, Mgr. Zwijsen College te Veghel
2 juli 1987 doctoraal examen wiskunde (met lof), Technische Universiteit Eindhoven
van 21 november 1986 deeltijdsleraar wiskunde, tot 1 augustus 1991 Hogeschool Eindhoven
vanaf 1 september 1987 assistent in opleiding, Technische Universiteit Eindhoven
143
STELLINGEN
behorende bij het proefschrift
INTEGRAL TRANSFORMATIONS AND
SPACES OF TYPES
door C.A.M. van Berkel
1. Beschouw de recurrente betrekking
(n + 1) (2n + 1) Cj,n+I = c;-t,n + 2n2 Cj,n + n(2n + 1) c;,n-1
-2n(n -1)c;,n-2 , 0 :5 j :5 n + 1, nE !No,
met randwa.arden ~.o = 1 , Cj,n = 0 als j < 0 of j > n.
Voor n E JN0 geldt
.. L: Cj,n = 1 en Cj,n ?; 0 , 0 :5 j :5 n . j=O
2. Voor p E 1R en n E JN0 geldt
(x1- 11 d/dx)2"+1 = (x 1- 2Pdfdx)" x11(2n+l) (x1- 211 d/dx)"+1 •
3. Voor t E JR+ en n, m, q E JN0 met q ?: 2 geldt
t"' e-t !L:'+q/2-t(tW
1rlll2 f(n +m+ q/2) f(2n +m+ q) f(m + 1) :5 (q- 1) (2m + q- 2) f(q/2) f(m + q- 2) r(2n +m+ 1) f(n + 1) .
Hierin is L~ het gegeneraliseerde Laguerrepolynoom,
e"' d" L"(x) = x-v - - [e-"' x"+"] .
"' n! dx"
(Vergelijk Duran [lJ)
4. Zij 11 E C met Re 11 > 0 en zij K,.. de verzameling van alle functies van de vorm
t ~--+ (f?(l- 2 tanh2 t) cosh-" t,
met (f? een geheel analytische functie. Het beeld van K,.. onder de Fouriertransformatie is de verzameling van alle functies van de vorm
x ~--+ f((11 + ix)/2) f((11- ix)/2) '1/J(x),
met '1/J een even en geheel analytische functie van sub-exponenWHe groei, i.e.
'rf~>O SUp exp(-elzl) 1'1/J(z)l < 00. zec
(Van Berkel en De Gra.af [2])
5. Zij L: het gegeneraliseerde Laguerrepolynoom als in Stelling 3. Definieer voor n, m E No de functie U,.,m op .112 door
Dan wordt de Radongetransformeerde 'RU,.,m gegeven door
p E .11 , w E .112 met w~ + w~ = 1 .
Hierin is H e1c het Hermitepolynoom,
6. Zij f een snel afnemende functie op JR'l en veronderstel dat haar Radongetransformeerde 'Rf de eigenschap heeft dat ('R.f) (p,w) = 0 voor IPI > a. Dan is f(x) = 0 voor lxl >a. Deze stelling kan eenvoudig bewezen worden met behulp van de Weyl-Gegenbauertransformaties en ontwikkelingen naar spherische harmonischen.
(Voor andere bewijzen zie Helgason 13] of Ludwig 14])
7. Zij Pm een homogeen harmonisch polynoom van de graad m op Bq. Dan geldt voor lE lNJ en e E S9-
1 ={wE JR'l : wi + ... +w; = 1},
l(a'pm) {e)l2 :5 A!,. J 1Pml2 dt7q-l
Srl
met
1 11'1/2-q/2 211l-q+2 f(m + 1) f(m + q/2) r(m Ill+ q- 2) l - . m- f(q/2 -1/2) If( m -Ill+ 1)]2 r(m Ill+ q/2- 1)
Hierin is 81 = 811 + ... +l, / (8x~1 ••• 8x~') en qq-t de oppervlaktemaat zo dat qq-t(Sq.,.t) = 21rq/2j f(q/2).
(Van Berkel en Van Eijndhoven [5])
8. Zij F de Fresnelintegraal gedefinieerd door
00
F(z) = j exp(it2 )dt. z
Voor x > 0 geldt
J e~:~~;~) dt = 2i:n- e'"/4 FJ2 e-2iz F(../2X) . 0
Met behulp van dit resultaat is de integraal /(.\) in Servadio [6, Appendix B] uit te drukken in een Fresnelintegraal, waarna de asymptotische ontwikkeling van /(.\) voor ,\-+ oo eenvoudig volgt uit die van de Fresnelintegraal.
9. Zij Jv de Besselfunctie van de eerste soort en van orde 11. Voor m, n E JN0 en 0 < x :::; 1r geldt
1
f: k-l Jm+t(kx) Jn+!.(kx) = hm,nl •
k=-oo " 2 n + 2
Deze betrekking is te beschouwen als een discrete versie van de orthogonaliteitsbetrekking
Literatuur
[1] A.J. DURAN, A bound on the Laguerre polynomials. Studia Math. 100, 169-181 (1991).
[2] C.A.M. VAN BERKEL and J. DE GRAAF, On a property of the Fourier-cosine transform. Appl. Math. Lett. 1, 307-310 (1988).
[3] S. HELGASON, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Gra.ssma.nn manifolds. Acta Math. 113, 153-180 (1965).
[4] D. LUDWIG, The Radon transform on Euclidean space. Comm. Pure Appl. Math. 19, 49-81 (1966).
[5] C.A.M. VAN BERKEL and S.J.L. VAN EIJNDHOVEN, Estimates for spherical harmonics. RANA 89-20, Eindhoven University of Technology (1989).
[6] S. SERVADIO, Flux integrals for Sommerfeld's half-plane problem. Phys. Rev. A 24, 793-800 (1981).