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Multidimensional Chebyshev spaces,hierarchy of innite-dimensional spaces and

Kolmogorov-Gelfand widths

O. KounchevInstitute of Mathematics and Informatics,

Bulgarian Academy of Sciencesand

IZKS, University of Bonn

May 9, 2011

Abstract

Recently the theory of widths of Kolmogorov (especially of Gelfand

widths) has received a great deal of interest due to its close relation-ship with the newly born area of Compressed Sensing. It has beenrealized that widths reect properly the sparsity of the data in SignalProcessing. However fundamental problems of the theory of widthsin multidimensional Theory of Functions remain untouched, and theirprogress will have a major impact over analogous problems in the the-ory of multidimensional Signal Analysis. The present paper has threemajor contributions:

1. We solve the longstanding problem of nding multidimensionalgeneralization of the Chebyshev systems: we introduce Multidimen-sional Chebyshev spaces, based on solutions of higher order elliptic

equation, as a generalization of the one-dimensional Chebyshev sys-tems, more precisely of the ECTsystems.

2. Based on that we introduce a new hierarchy of innite-dimensionalspaces for functions dened in multidimensional domains; we denecorresponding generalization of Kolmogorovs widths.

3. We generalize the original results of Kolmogorov by computingthe widths for special "ellipsoidal" sets of functions dened in multi-dimensional domains.

MSC2010: 35J, 46L, 41A, 42B.

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Keywords: Kolmogorov widths, Gelfand widths, Compressed Sens-

ing, Chebyshev systems, Approximation by solutions of PDEs.

1 Introduction

It is a notorious fact that the polynomials of several variables fail to enjoythe nice interpolation and approximation properties of the one-dimensionalpolynomials, and this is particularly visible in such fundamental areas ofMathematical Analysis as the Moment problem, Interpolation, Approxima-tion, etc. One alternative approach is to use solutions of elliptic equations, inparticular polyharmonic functions, which has led to new amazing Ansatzes inmultidimensional Approximation and Interpolation [17], [15], [13], [24], [18],in Spline and Wavelet Theory [32], [22], and recently in the Moment Problem[26]. This approach has been given the name Polyharmonic Paradigm [22], asan approach to Multidimensional Mathematical Analysis, which is oppositeto the usual concept which is based on algebraic and trigonometric poly-nomials of several variables. However the eectiveness of the PolyharmonicParadigm has remained unexplained for a long time.

One of the main objectives of the present research is to nd a new point ofview on the longstanding problem of nding multidimensional generalizationof Chebyshev systems: In Section 2 we show that the solution spaces of a wide

class of elliptic PDEs (dened further as Multidimensional Chebyshev spaces)are a natural generalization of the one-variable polynomials as well as of theone-dimensional Chebyshev systems.1 In particular, this shows that the poly-harmonic functions which are solutions to the polyharmonic operators are ageneralization of the one-variable polynomials. Let us recall that there hasbeen a long search for proper multidimensional generalization of Chebyshevsystems. The standard generalization by means of zero set property fails toproduce a non-trivial multidimensional system which is the content of thetheorem of Mairhuber, see the thorough discussion in [29] (chapter 2; section1:1). Our generalization provided by the Multidimensional Chebyshev spaces

comes from a completely dierent perspective, by generalizing the boundaryvalue properties of the one-dimensional polynomials.2

1 Recall that the one-dimensional Chebyshev systems appeared as a generalization ofthe one-variable algebraic polynomials in the works of A. Markov in the context of theclassical Moment problem. They were further developed and applied to the generalizedMoment problem and Approximation theory by A. Haar (the Haar spaces), S. Bernstein,M. Krein, S. Karlin, and others, cf. [29], [20], [34].

2 In general, zero set properties and intersections are not a reliable reference pointfor multidimensional Analysis. In particular, let us recall that polyharmonic (and evenharmonic) functions do not have simple zero sets, although they are solutions to nice BVPs

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On the other hand the cornerstone of the present paper is an amaz-

ing though simple characterization of the Ndimensional subspaces XN ofCN1 (I) (here I is an interval in R) via Chebyshev systems. This charac-terization says that the typical subspace XN is a nite-piecewise Chebyshevspace, or to be more correct, nite-piecewise ECT-system. This discoverycauses an immediate chain reaction: by analogy, for a domain D Rn;we use the newly-invented Multidimensional Chebyshev spaces to dene inL2 (D) a multidimensional generalization XN of the spaces XN; which we call"spaces of Harmonic Dimension N". These spaces XN represent a new hier-archy of innite-dimensional spaces. Hence, the big surprise of the presentresearch is the reconsideration of the simplistic understanding that the nat-

ural generalization ofXN is provided just by the nite-dimensional subspacesofL2 (D) : We realize that the nite-dimensional subspaces in C

N (D) ; for do-mains D Rn for n 2; do not serve the same job as the nite-dimensionalsubspaces in CN (D) for intervals D R1; and one has to replace them bya lot more sophisticated objects, namely by the spaces having HarmonicDimension N:

Respectively, the focus of the present research is, by means of the spacesXN to dene a multidimensional generalization of the Kolmogorov-GelfandNwidths, which we call "Harmonic Nwidths". After that we compute theHarmonic Nwidths for "cylindrical ellipsoids" in L2 (D) ; by generalizingthe original results of Kolmogorov.Another important motivation for the present research is the recent in-terest to the theory of widths (especially to Gelfand widths) coming fromthe applications in an area of Signal Analysis, called Compressed Sensing(CS). In a certain sense the central idea of CS is rooted in the theory ofwidths, cf. e.g. [10], [8], [9], [37]. However, apparently this strategy workssmoothly only in the case of representation of one-dimensional signals, whilean adequate approach to multivariate signals is missing one reason may befound by analogy in the fact that the theory of Kolmogorov-Gelfand widthsts properly only for one-dimensional function spaces (as pointed out below,e.g. in formula (27)). Recently, a new multivariate Wavelet Analysis wasdeveloped based on solutions of elliptic partial dierential equations ([22]),in particular "polyharmonic subdivision wavelets" were introduced (cf. [11],[27]). In order to apply CS ideas to these wavelets it would require essentialgeneralization of the theory of widths for innite-dimensional spaces, and itis expected that the present research is a step in the right direction.

In his seminal paper [21] Kolmogorov has introduced the theory of widthsand has applied it ingeniously to the following set of functions dened in the

as (38)-(39), cf. [18].

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compact interval:

Kp :=

f 2 ACp1 ([a; b]) :

Z10

f(p) (t)2 dt 1 : (1)In the present paper we study a natural multivariate generalization of theset Kp which in a domain B Rn is given by

Kp :=

u 2 H2p (B) :

ZB

jpu (x)j2 dx 1

; (2)

where p

is the pth iterate of the Laplace operator =n

Xj=1

@2

=@x2j ; we

consider more general sets Kp given in (26) below.Let us summarize the major contributions of the present paper:

1. For every integer N 0 we dene the Multidimensional Chebyshevspaces of order N as spaces of solutions of a special class of elliptic PDEsof order 2N: They generalize the classical one-dimensional ExtendedComplete Chebyshev systems (ECTsystems).

2. For every integer N

0 we generalize the N

dimensional subspaces

XN in CN1 (I) (for intervals I R) to a multidimensional setting.The generalization XN is a piecewise Multidimensional Chebyshevspace of order N; and is said to have "Harmonic Dimension N". Thisrepresents a new hierarchy of innite-dimensional spaces of functionsdened in domains in Rn:

3. For every integer N 0 we dene Harmonic Widths which generalizethe Kolmogorov widths, whereby we use as approximants the spacesXN instead of nite-dimensional spaces XN used by the Kolmogorovwidths. We generalize the one-dimensional Kolomogorovs results inthe theory of widths.

The crux of the new notion of hierarchy of innite-dimensional spaces isthe following: Let the domain D Rn be compact with suciently smoothboundary @D: Then the Ndimensional subspaces in C1 (I) will be gener-alized by spaces ofsolutions of elliptic equations (and by more general spacesintroduced in Denition 14 below):

XN = fu : P2Nu (x) = 0; for x 2 Dg L2 (D) ; (3)

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here P2N is an elliptic operator of order 2N in the domain D: Respectively,

the simplest version of our generalization of Kolmogorovs theorem aboutwidths nds the extremizer of the following problem

infXN

distXN; Kp ;

where Kp is the set dened in (2) and XN is dened in (3) by an ellipticoperator P2N of order 2N; for the complete formulation see Theorem 23below.

What is the reason to take namely solutions of elliptic equations in themultidimensional case is explained in the following section.

Acknowledgements: The author acknowledges the support of theAlexander von Humboldt Foundation, and of Project DO-02-275 with Bul-garian NSF. The author thanks the following Professors: Matthias Leschfor the interesting discussion about hierarchies of innite-dimensional linearspaces, Hermann Render about advice on multivariate polynomial division,and Peter Popivanov, Nikolay Kutev and Georgi Boyadzhiev about adviceon Elliptic BVP.

2 Multidimensional Chebyshev spaces and a

hierarchy of innite-dimensional spacesLet us give a heuristic outline of the main idea of this new hierarchy ofspaces, by explaining how it appears as a natural generalization of the nite-dimensional subspaces of CN1 [a; b] in a compact interval [a; b] in R: Firstof all, we will show that there exists an amazing relation between the nite-dimensional subspaces of CN1 [a; b] and the theory of Chebyshev systems.

Let us construct some special type ofNdimensional subspaces of CN1 (J)where the interval J = (c; d) R: Let the functions j; j = 1; 2;:::;N satisfyj > 0 on J; and j 2 CN+1j (J) : We assume that the functions j satisfythe necessary integrability so that we may dene the following functions:

v1 (t) = 1 (t) (4)

v2 (t) = 1 (t)

Ztc

2 (t2) dt2 (5)

vN (t) = 1 (t)

Ztc

2 (t2)

Zt2c

3 (t3) ZtN1

c

N (tN) dt2dt3 dtN: (6)

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Let us recall some classical results about the space XN = span

fvj

gNj=1 :

For every k = 1; 2;:::;N the consecutive Wronskians for the system of func-tions fvjgNj=1 may be computed explicitly and are given by

Wk := W(v1; v2;:::;vk) = k1

k12 k; (7)

and vice versa:

1 = W1 = v1; 2 = W2=W21 (8)

k = WkWk2=W2k1 for k 3; (9)

(cf. [38], or [20], chapter 11; formulas 1:12 and 1:13). From representationformula (7) directly follows that for all k = 1; 2;:::;N the Wronskians satisfy

W(v1; v2;:::;vk) > 0 on J: (10)

Let us dene on J the ordinary dierential operator

LN

t;

d

dt

=

d

dt

1

N (t) d

dt

1

2 (t)

d

dt

1

1 (t); (11)

Then, from formulas (4)(6) directly follows that all vjs, hence all elements

of the space span fvjgN

j=1 ; satisfy the ODE

LNu (t) = 0; for t 2 J: (12)

Obviously, the operator LN has a non-negative leading coecient and is inthis sense one-dimensional "elliptic".

We have the following classical result (cf. [20], chapter 11; Theorem 1:2).

Proposition 1 The space

XN = span fvjgNj=1 (13)

is an Ndimensional subspace of CN1 (J) :

We recall the following denition (cf. [20], chapter 11).

Denition 2 A space XN CN1 (J) is called ECTspace (or ExtendedComplete Chebyshev space) if it has a basis fvjgNj=1 satisfying the Wron-skian condition (10).

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Our terminology above diers slightly from the accepted terminology: we

consider Chebyshev systems on open intervals J and instead of "ECT-system"we say "ECT-space".

The following result characterizes the typical ("general position") Ndimensionalsubspaces of CN1 [a; b] by means of the ECT-spaces.

Theorem 3 "Almost all"Ndimensional subspaces ofCN1 [a; b] are nite-piecewise ECTspaces in the following sense: If XN is an Ndimensionalsubspace of CN1 [a; b] then there exists a sequence of Ndimensional sub-spaces XmN of C

N1 [a; b] ; m 1; satisfying:1. For every space XmN there exists a nite subdivision a t0 < t1

0; will be denoted by fjg1j=1 : The set of functions fjg1j=1S

0j1j=1

form

a complete orthonormal system in L2 (B) :

Remark 11 Problem (29)-(30) is well known to be a non-regular ellipticBVP, as well as non-coercive variational, cf. [1] ( p. 150 ) and [30] (Remark9:8 in chapter 2; section 9:6, and section 9:8 ).

The proof is provided in the Appendix below, section 11.1.

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6 The principal axes of the ellipsoid Kp anda Jackson type theorem

Here we will nd the principal exes of the ellipsoid Kp dened as

Kp :=

u 2 H2p (B) :

ZB

jL2pu (x)j2 dx 1

; (31)

where L2p is a uniformly strongly elliptic operator in B:We prove the following theorem which generalizes Kolmogorovs one-

dimensional result from Proposition 6, about the representation of the el-

lipsoid Kp in principal axes.

Theorem 12 Let f 2 Kp : Then f is represented in a L2series as

f(x) =1X

j=1

f0j0j (x) +

1Xj=1

fjj (x) ;

where by Theorem 10 the eigenfunctions 0j satisfy p0j (x) = 0 while the

eigenfunctions j correspond to the eigenvalues j > 0; and also

1Xj=1

jf2j 1: (32)

Vice versa, every sequence

f0j1j=1

S ffjg1j=1 with 1Xj=1

f0j2 + 1Xj=1

jfjj2 < 1

and1X

j=1

jf2j 1 denes a function f 2 L2 (B) which is in Kp :

Proof. (1) According to Theorem 10, we know that arbitrary f 2 L2 (B) isrepresented as

f(x) =1X

j=1

f0j0j (x) +

1Xj=1

fjj (x)

kfk2L2 =1X

j=1

f0j2 + 1Xj=1

jfj j2 < 1

with convergence in the space L2 (B) :

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(2) From the proof of Theorem 10, we know that if we put

j (x) = L2pj (x) for j 1;then the system of functions

j (x)pj

for j 1

is orthonormal sequence which is complete in L2 (B) :(3) We will prove now that if f 2 L2 (B) then f 2 Kp i

1

Xj=1

f2

j j 1:

Indeed, for every f 2 H2p (B) we have the expansion f(x) =1X

j=1

f0j0j (x) +

1Xj=1

fjj (x) : We want to see that it is possible to dierentiate termwise this

expansion, i.e.

L2pf(x) =1

Xj=1fjL2pj (x) =1

Xj=1fjj (x)Since

jpj

j1

is a complete orthonormal basis of L2 (B) it is sucient to

see that ZB

L2pf(x) jdx =

ZB

1X

j=1

fjL2pj (x)

!jdx:

Due to the boundary properties of j and since j = L2pj; we obtain

ZBL2pf(x) jdx = ZBf(x) L2pjdx = jZBf jdx = jfj:

On the other hand ZB

1X

k=1

fkk (x)

!jdx = jfj:

Hence

L2pf(x) =1X

j=1

fjL2pj (x) =1X

j=1

fjj (x) =1X

j=1

pjfj

j (x)pj

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and since jpjj1 is an orthonormal system, it followskL2pfk2L2 =

1Xj=1

jf2j :

Thus if f 2 Kp it follows that1X

j=1

jf2j 1:

Now, assume vice versa, that1X

j=1

f2j j 1 holds together with1X

j=1

f0j2

+

1Xj=1

jfjj2 < 1. We have to see that the function

f(x) =1X

j=1

f0j0j (x) +

1Xj=1

fjj (x)

belongs to the space H2p (B) : Based on the completeness and orthonormality

of the system

j(x)p

j

1j=1

we may dene the function g 2 L2 by putting

g (x) =

1Xj=1

pjfj j (x)pj

=

1Xj=1

fjj (x) ;

it obviously satises kgkL2 1:From the local solvability of elliptic equations ([30]) there exists a function

F 2 H2p (B) which is a solution to equation L2pF = g: Let its representationbe

F(x) =1X

j=1

f0j0j (x) +

1Xj=1

Fjj (x)

with some coecients Fj satisfying Xj jFjj2

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