Fourier Analysis and Approximation of Functions

594 Pages·2016·43.43 MB·English

by  Roald M. Trigub and Eduard S. Bellinsky

Fourier Analysis and Approximation of Functions
Fourier Analysis
and Approximation of
Functions
by
Roald M. Trigub
Donetsk National University,
Donetsk, Ukraine
and
Eduard S. Bellinsky
University of West Indies,
Bridgetown, Barbados
Springer Science+Business Media, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-6641-1 ISBN 978-1-4020-2876-2 (eBook)
DOI 10.1007/978-1-4020-2876-2
Printed on acid-free paper
All Rights Reserved
© 2004 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2004.
Softcover reprint of the hardcover 1s t edition 2004
No part of this work may be reproduced, stored in a retrieval system, or transmitted
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DEDICATED TO
THE MEMORY OF
OUR PARENTS
Contents
Dedication v
Preface Xl
1. REPRESENTATION THEOREMS 1
1.1 Theorems on representation at a point 1
1.2 Integral operators. Convergence in Lp-norm and almost
everywhere 5
1.3 Multidimensional case 16
1.4 Further problems and theorems 20
1.5 Comments to Chapter 1 24
2. FOURIER SERIES 25
2.1 Convergence and divergence 25
2.2 Two classical summability methods 34
2.3 Harmonic functions and functions analytic in the disk 42
2.4 Multidimensional case 50
2.5 Further problems and theorems 59
2.6 Comments to Chapter 2 66
3. FOURIER INTEGRAL 67
3.1 L-Theory 67
3.2 L2-Theory 75
3.3 Multidimensional case 80
3.4 Entire functions of exponential type. The Paley-Wiener
theorem 88
3.5 Further problems and theorems 95
3.6 Comments to Chapter 3 104
Vlll FOURIER ANALYSIS AND APPROXIMATION OF FUNCTIONS
4. DISCRETIZATION. DIRECT AND INVERSE THEOREMS 105
4.1 Summation formulas of Poisson and Euler-Maclaurin 106
4.2 Entire functions of exponential type and polynomials 113
4.3 Network norms. Inequalities of different metrics 125
4.4 Direct theorems of Approximation Theory 132
4.5 Inverse theorems. Constructive characteristics. Embedding
theorems 138
4.6 Moduli of smoothness 144
4.7 Approximation on an interval 169
4.8 Further problems and theorems 189
4.9 Comments to Chapter 4 196
5. EXTREMAL PROBLEMS OF APPROXIMATION THEORY 201
5.1 Best approximation 201
5.2 The space Lp. Best approximation 205
5.3 Space C. The Chebyshev alternation 212
5.4 Extremal properties for algebraic polynomials and splines 218
5.5 Best approximation of a set by another set 234
5.6 Further problems and theorems 245
5.7 Comments to Chapter 5 252
6. A FUNCTION AS THE FOURIER TRANSFORM OF
A MEASURE 255
6.1 Algebras A and B. The Wiener Tauberian theorem 256
6.2 Positive definite and completely monotone functions 264
6.3 Positive definite functions depending only on a norm 278
6.4 Sufficient conditions for belonging to Ap and A * 288
6.5 Further problems and theorems 304
6.6 Comments to Chapter 6 307
7. FOURIER MULTIPLIERS 309
7.1 General properties 309
7.2 Sufficient conditions 321
7.3 Multipliers of power series in the Hardy spaces 329
7.4 Multipliers and comparison of summability methods of
orthogonal series 337
7.5 Further problems and theorems 345
Contents IX
7.6 Comments to Chapter 7 347
8. SUMMABILITY METHODS. MODULI OF SMOOTHNESS 349
8.1 Regularity 349
8.2 Applications of comparison. Two-sided estimates 358
8.3 Moduli of smoothness and K-functionals 368
8.4 Moduli of smoothness and strong summability in Hp(D),
0< p::; 1 375
8.5 Further problems and theorems 385
8.6 Comments to Chapter 8 389
9. LEBESGUE CONSTANTS AND APPROXIMATION 393
9.1 Upper and lower estimates 394
9.2 Examples of Lebesgue constants in the multiple case 404
9.3 Asymptotics of Lebesgue constants and approximation 410
9.4 Further problems and theorems 423
9.5 Comments to Chapter 9 427
10. WIDTHS. POLYNOMIAL APPROXIMATION 429
10.1 Entropy numbers 429
10.2 Polynomials with free spectrum. Trigonometric widths 438
10.3 Kolmogorov widths 450
10.4 Further problems and theorems 469
10.5 Comments to Chapter 10 475
11. FUNCTIONS WITH BOUNDED MIXED DERIVATIVE 477
11.1 Hyperbolic cross polynomials 477
11.2 Estimates of entropy numbers 486
11.3 Widths 499
11.4 Further problems and theorems 506
11.5 Comments to Chapter 11 509
Appendices 511
A Prerequisites 511
A.l Weierstrass approximation theorems 511
A.2 The modulus of continuity of a function 512
A.3 The Riemann-Stieltjes integral 513
A.4 Summability of series 514
x FOURIER ANALYSIS AND APPROXIMATION OF FUNCTIONS
A.5 Analytic functions 516
A.6 Measure and integral. Complex-valued measures 519
A.7 Hilbert spaces. Classical orthonormal systems 525
A.8 Banach spaces and linear operators 531
A.9 Fourier multipliers 534
A.lO Several classical theorems 536
A.l1 Polynomials with integral coefficients 537
A.12 Some inequalities 537
B Principal symbols 541
B.1 ~m and its subsets 541
B.2 Functional notations 541
References 543
Author Index 577
Topic Index 588
Preface
In this book basics of classical Fourier Analysis are given (Fourier
series and integrals, the Fourier transform of a measure, etc.) as well as
those of approximation by polynomials, splines, and entire functions of
exponential type (direct and inverse theorems, approximation properties
of summability methods of simple and multiple Fourier series, extremal
problems).
Our interest in special results in Fourier Analysis is mainly motivated
by certain approximation problems, albeit many of the obtained results
in Fourier Analysis are of considerable importance in their own right.
Nevertheless, with almost no exclusion such results are applied to prov
ing one or another problem in Approximation Theory.
In Chapter 1, which is of introductory nature, theorems on conver
gence, in that or another sense, of integral operators are given. In Chap
ter 2 basic properties of simple and multiple Fourier series are discussed,
while in Chapter 3 those of Fourier integrals are studied.
The first three chapters as well as partially Chapter 4 and classi
cal Wiener, Bochner, Bernstein, Khintchin, and Beuding theorems in
Chapter 6 might be interesting and available to all familiar with fun
damentals of integration theory and elements of Complex Analysis and
Operator Theory. Applied mathematicians interested in harmonic anal
ysis and/or numerical methods based on ideas of Approximation Theory
are among them. The book is supplied with appendix in which theorems
from general mathematical courses as well as some special results used
throughout the text are collected.
When writing the rest of the book, the authors followed the idea that
one of the main goals of writing a monograph is a sort of summarizing
and stimulating further study of the problems discussed. In Chapters 6-
11 very recent results are sometimes given in certain directions. These
are multipliers of Fourier series and multipliers of power series in the