Uniform Approximations by Trigonometric Polynomials [Reprint 2018 ed.] 9783110926033, 9783110460773

Citation preview

Uniform Approximations by Trigonometric Polynomials

Uniform Approximations by Trigonometric Polynomials

A.I. Stepanets Translated by V.V. Gorunovich P.V. Malyshev D.V. Malyshev

MVS?Ill Utrecht • Boston • Koln • Tokyo, 2001

VSP BV P.O. Box 346 3700 AH Zeist The Netherlands

Tel: +31 30 692 5790 Fax:+3 1 30 693 2081 [email protected] www.vsppub.com

© V S P BV 2001 English edition first published in 2001 Russian edition first published in 1981 ISBN 90-6764-347-5

Translated by V.V. Gorunovich*, P.V. Malyshev and D.V. Malyshev t

The work on the translation was in progress when one of the translators, our friend V.V. Gorunovich passed away in an accident. We dedicate our work to his bright memory - P.M., D.M.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Printed in The Netherlands by Ridderprint bv, Ridderkerk.

CONTENTS

Preface to the English Edition

ix

Preface to the Russian Edition

xi

Introduction

1

Chapter 1. SIMPLEST EXTREMAL PROBLEMS

7

1. Modulus of Continuity

7

2. Classes of Continuous Functions

11

3. Simplest Extremal Problem for the Classes

b]

14

4. Scheme of Estimation of One-Dimensional Integrals

24

5. Simplest Extremal Problem for the Classes H%\P)

38

6. Representation of ^ ( y , cp; H ^ \ P ) ) in Terms of Rearrangements

61

7. Case of Symmetric Functions

68

8. Simplest Extremal Problem for the Classes H^\P),

N> 2

Bibliographical Notes

78 97

Chapter 2. APPROXIMATION OF FUNCTIONS OF ONE VARIABLE BY FOURIER SUMS 1. Fourier Sums

99 99

2. Conjugate Functions and Their Classes

10S

3. Asymptotic Relations for Zeros of Integral Sine and Integral Cosine

108

4. Representations of Upper Bounds of Deviations of Fourier Sums on the Classes WTHm

122

5. Asymptotic Estimates for %

130

Sn)

vi

Contents 6. Asymptotic Estimates for % (WrHw; S„)

142

7. Simultaneous Approximation of Periodic Functions and Their Derivatives by Fourier Sums

152

8. Asymptotic Estimates for %(Ha;Sn)

157

9. Asymptotic Estimates for %{WrH^\ Sn)

165

10. Simultaneous Approximation of Derivatives of Functions from the Class WrHm by Fourier Sums

169

Bibliographical Notes

172

Chapter 3. APPROXIMATION OF FUNCTIONS OF MANY VARIABLES BY FOURIER SUMS

179

1. Multiple Fourier Sums. Statement of the Problem

179

2. Decomposition of Functions into Simple Functions

188

3. Dirichlet Integral for Simple Functions

195

4. Estimates for Deviations of Fourier Sums on the Class H ^

201

5. Extremal Functions and Asymptotic Equalities

210

Bibliographical Notes

227

Chapter 4. FEJERSUMS

235

1. Linear Methods of Summation of Fourier Series. General Aspects

235

2. Bernoulli Kernels and Extremal Values of Periodic Functions

247

3. Deviation of Fejir Sums on the Classes W H ^

260

4. Fejir Sums on Classes of Conjugate Functions

277

5. Deviation of Fej£r Sums on the Classes H^

294

Bibliographical Notes

297

Chapter 5. SPHERICAL RIESZ SUMS

301

1. Bochner Formula 2. Deviation of the Riesz Sums Sj,(/;x)

301 on the Classes W / / ^

309

3. Asymptotic Equalities for %sn(WrHl0)

327

4. Representation of ^ ( H ^ ) and Auxiliary Statements

337

Contents

vii

5. Estimates from Above for

344

6. Asymptotic Equalities for

349

Bibliographical Notes

353

Chapter 6. ROGOSINSKI SUMS

355

1. Definitions and General Remarks

353

2. Distribution of Zeros of Functions Associated with Rogosinski Sums

359

3. Estimates from Above for

372

Rn)

4. Behavior of the Quantities %(WrHai\ Rn)

378

5. Rogosinski Sums on the Classes H ^

387

Bibliographical Notes Chapter 7. FAVARD SUMS

396 399

1. Definitions and Auxiliary Remarks

399

2. Location of Zeros of the Favard Kernel

404

3. Location of Zeros of the First Derivative of the Favard Kernel

414

4. Location of Zeros of the Second Derivative of the Favard Kernel

422

5. Estimates for the Lebesgue Constant of the Favard Method

435

6. Properties of the Function p*(f)

442

7. Favard Sums on the Classes H a

446

8. Favard Sums on the Classes H ^

449

Bibliographical Notes

467

REFERENCES

469

INDEX

483

PREFACE TO THE ENGLISH EDITION

The material presented in this monograph did not lose its importance since the publication of the Russian edition and is undoubtedly of great interest to Western readers. This is explained by the fact that this field of mathematics was mainly developed in the Soviet Union under the influence of prominent mathematicians of the 20th century such as A. N. Kolmogorov, S. M. Nikol'skii, S. B. Stechkin, V. K. Dzyadyk, N. P. Komeichuk, etc., and, for well-known reasons, the results obtained by Soviet mathematicians are less known in the West. It should be noted that, in this field of mathematics, numerous deep results were obtained, many important complicated problems were solved, and new methods were developed, which can be extremely useful for many mathematicians. It should also be noted that the solutions of all principal problems considered in this book are given in the final form, i.e., in the form of exact asymptotic equalities, and, therefore, they retain their importance and interest for a long time. As compared with the Russian edition, a new chapter (Chapter 7) has been added and several misprints have been fixed. A. Stepanets

PREFACE TO THE RUSSIAN EDITION

The theory of approximation of functions is one of the central branches of the mathematical analysis. This theory appeared as a result of the internal development of mathematics and requirements of practice and it has been intensively developed for many decades. In this theory, in terms of the concept of function, one of the fundamental ideas of the mathematics, namely, the approximation (replacement) of complex objects by simpler and more convenient ones, is reflected. This idea is principal for problems concerning the relationship between mathematics and practice, which stimulated the development of the theory of approximation of functions in the past and will obviously guarantee an interest in it in the future. In the present book, we consider a series of problems related to one of directions of the theory indicated, namely, to the approximation of periodic functions by trigonometric polynomials generated by linear methods of summation of Fourier series. On the other hand, this direction can also be related to the theory of Fourier series. Due to this, the results of this book are located at the intersection of both mentioned theories supplementing one another. At present, the investigations related to linear methods of summation of Fourier series form an entire field of mathematics. In this field, there has been a series of statements of problems and many deep results have been obtained, and it is impossible to sketch them in a single book. In view of this, we restrict ourselves to the investigation of specific linear methods, namely, the classical methods of Fourier, Fej£r, Riesz, and Rogosinski. For these methods, we mainly consider the so-called Kolmogorov-Nikol'skii problem. It consists of finding exact and asymptotically exact equalities for the upper bounds of deviations of polynomials generated by given linear methods on given classes of 27i-periodic functions. As such classes, we consider the classes of continuous and differentiate functions, and the deviations are measured in the uniform metric. Much attention is given to the multidimensional case. In particular, the KolmogorovNikol'skii problem is solved on classes of continuous functions for multiple rectangular Fourier sums and spherical Riesz sums. In the first chapter, an auxiliary material is given and general schemes are presented for estimates of upper bounds of integrals on classes of continuous functions, which are based on the well-known Komeichuk-Stechkin lemma and its multidimensional analogs. The contents of the other chapters are clear from their titles.

xii

Preface to the Russian Edition

When presenting the main material, as a rule, we do not refer the reader to the literature in the core text. All references are given in the bibliographical notes at the end of each chapter. As a rule, all statements are presented with detailed proofs, except certain auxiliary facts whose detailed proofs can be found in standard university courses in analysis or in textbooks on trigonometric series and Bessel functions. The chapters of the book are split into sections, which, in turn, are split into subsections enumerated by two numbers: the first stands for the number of the section and the second for the number of the subsection itself. The same enumeration is used for all kinds of statements and formulas. If we refer to statements or formulas from a different chapter, we use triple enumeration, where the first number stands for the chapter and the other two have the meaning described above. The new results presented in this book were discussed at seminars held at the Institute of Mathematics of the Ukrainian Academy of Sciences and the Steklov Mathematical Institute of the Academy of Sciences of the USSR. I am deeply grateful to all participants of these seminars who took an active part in discussions. The present book was written on the basis of many-year investigations initiated in author's thesis. Professor Dzyadyk, the teacher of the author, not only formulated the problem of finding asymptotic equalities for upper bounds of deviations of Rogosinski sums on the classes WrH", but also helped to overcome the difficulties appeared in the course of its solution. I am grateful to him and to Professors N. P. Korneichuk and S. B. Stechkin, whom I also regard as my teachers. Their ideas, methods, and attention played an important role in my scientific carrier. I am also grateful to Professors N. P. Korneichuk and S. B. Stechkin for reading the manuscript of this book and helpful comments. I am also grateful to my disciple P. V. Zaderei for careful proofreading of the manuscript of this monograph. I am deeply grateful to my wife and colleague N. I. Stepanets for the laborious work of preparing the manuscript for publishing. Without her help, it would take much more time to publish the monograph. A. Stepanets

INTRODUCTION

Let /(JE) be a 27t-periodic continuous (or summable) function, let

— + 2

(ak cos kx+bk sin kx) k=l

be its Fourier series, let 1 * ak = a k ( f ) = - J f 71 - I t

f(t)cosktdt,

1 * h = W ) = ~ f /(OsinJfciA, —71

k =

0,1,...,

be its Fourier coefficients, and let

£„(/;*)=

2

+X

*=i

(akcoskx+bksmkx)

be the partial Fourier sum of the nth order of the Fourier series (the Fourier sum). The first results on the estimation of the deviations of Fourier sums from given continuous functions were obtained in the course of development of the theory of approximation of functions. In 1909, Lebesgue [Lebl] proved that P„(/; *) =f |/(*)- Snif-, *)| < [Inn + 3] £„(/),

(1)

where En(f) is the best approximation of a function f(x) by trigonometric polynomials Tn(x) of degree not higher than n in the uniform metric:

2

Introduction

£„(/) = inf | | / W - r n W | | c 2K = inf max | / ( * ) - r „ ( x ) | . Tn( x)

Tn(x)

x

Together with the Jackson theorems on estimates for the quantities En(f) (see, e.g., [Kor6]), the Lebesgue inequality contains a major part of the early results on estimates for the quantities p„(/; x) and is of interest even today because it is exact in order and convenient for applications. For example, if a function f ( x ) has a derivative of the rth order bounded, e.g., by unity, then

and relation (1) yields ,

n(lnn + 3)

In 1935, Kolmogorov [Kol] considered the following quantity: %(Wr-,Sn)

where

Wr

= sup | | / W - 5 n ( / ; x ) | | c . feWr

is the class of 2ji-periodic functions

f(x)

whose rth derivatives

f^r\x)

(r is an integer, r > l ) satisfy the condition | / ( r ) ( x ) | < l almost everywhere. Kolmogorov showed that

it

n

\n

)

i.e., he obtained an asymptotically exact equality for the quantities % (Wr; Sn). The further progress in this direction was due to Nikol'skii [Nikl -Nik7], who generalized these results to the classes WrHa of 27t-periodic functions f(x)

having continu-

ous derivatives up to the rth order ( r £ 0 ) inclusive for which |/(r)W - / ( , ) 0 0 | ^ |*-*T.

0 < a < 1,

and to more general classes WrH(a determined by convex moduli of continuity oa(f). In particular, Nikol'skii established that, for any numbers r and a, r t 0, 0 < a < 1, the following equality is true:

Introduction

IT W

•

tasintdt

3 + o ( ^ ) .

Furthermore, he solved an analogous problem in the case where the Fejér sums o„(/; x) are considered instead of Sn(f ; x). These investigations of Kolmogorov and Nikol'skii initiated a new direction in the theory of approximation of functions and theory of summation of Fourier series. Later, the results of these investigations were extended to more general classes of functions, and trigonometric polynomials Un(J; x) generated by various methods Un of summation of Fourier series were used as approximation aggregates. The problem of finding asymptotic equalities for the quantities %mUH)=

sup /eSK

\\f(x)-Un(f;x)\\ C2,'

where Tl is a fixed class of continuous functions, became one of the most important problems in the theories of approximation of functions and summation of Fourier series. We call this problem the Kolmogorov-Nikol'skii problem. If an asymptotically exact equality is obtained for the quantities % (2ft; U„), i.e., a function ( 0 | < cd(|Ar|).

(4 )

On the other hand, by virtue of the uniform continuity of functions continuous on a closed interval, we have lim co(f) = 0. r-»+0 Thus, the function co(f) is right-continuous at the point ? = 0, which, together with (4'), proves its continuity for all r e [0, b - a]. 1.2. It follows from the property of semiadditivity of a modulus of continuity co(f) that to(ni) < nco(r) for any natural n, and

(5)

Section 1

Modulus of Continuity

9

0, (X + 1) t e [0, b - a]. Indeed, in the case n = 1, relation (5) is obvious. Assume that it is true for n = k, k^l. Then, according to (2), we have ©((Jk+1)0 = co(Jtf+/) < a(kt)+(o(t)

< (Jfc + l)©(f),

i.e., in this case, relation (5) is also valid for n = k + 1. Hence, relation (5) is true for any natural n. If X is an arbitrary positive number, then, denoting its integer part by [X] and taking into account relation (5) and the monotonicity of the function co(/), we obtain CD(Xf) < co([X] + l ) i < ([X]+ 1 )co(f) £ (X+l)oo(f). 1.3. If a function 9 ( f ) possesses properties (i)-(iv) on the interval [0, b - a], then the modulus of continuity 0)(])

if its modulus of continuity co(/; t) satisfies the following condition:

(o(f;t)