Commutative Harmonic Analysis IV: Harmonic Analysis in IRn

235 Pages·1992·6.88 MB·English

by  Sh. A. Alimov, R. R. Ashurov, A. K. Pulatov (auth.), V. P. Khavin, N. K. Nikol’skiǐ (eds.)

Encyclopaedia of
Mathematical Sciences
Volume42
Editor-in-Chief: R v. Gamkrelidze
V. P. Khavin N. K. Nikol'skii (Bds.)
Commutative
Harmonie Analysis IV
Harmonie Analysis in IRn
Springer-Verlag Berlin Heidelberg GmbH
Consulting Editors of the Series:
AAAgrachev, AAGonchar, E.P. Mishchenko, N.M.Ostianu,
V. P. Sakharova, AB. Zhishchenko
Title of the Russian edition:
Itogi nauki i tekhniki, Sovremennye problemy matematiki,
Fundamental'nye napravleniya, Vol. 42, Kommutativnyi garmonicheskii analiz 4
Publisher VINITI, Moscow 1989
Mathematics Subject Classification (1991):
31-xx, 42-xx, 43-xx, 46-XX
ISBN 978-3-642-08103-3 ISBN 978-3-662-06301-9 (eBook)
DOI 10.1007/978-3-662-06301-9
This work is subject to copyright. All rights are reserved, whether the whole or part ofthe material is concerned,
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© Springer-Verlag Berlin Heidelberg 1992
Originally published by Springer-Verlag Berlin Heidelberg New York in 1992.
Softcover reprint of the hardcover I st edition 1992
41/3140-543210· Printed on acid-free paper
Preface
In this volume of the series "Commutative Harmonie Analysis", three
points mentioned in the preface to the first volume are realized: 1) Multiple
Fourier series and Fourier integrals; 2) The machinery of singular integrals;
3) Exeeptional sets in harmonie analysis.
The first theme is the subjeet matter of the eontribution by Sh. A. Ali-
mov, R. R. Ashurov, A. K. Pulatov, whieh in an obvious way eonstitutes the
"multidimensional parallel" to S. V. Kislyakov's article in Volume I, devoted
to the "inner" questions of Fourier analysis of funetions of one variable. The
passage to the analysis of functions defined on ]Rn, n > 1, teIls us something
essential about the nature of the problem under study.
The eontribution by E. M. Dyn'kin, the beginning of which was already
published in Volume I of this subseries, is devoted to singular integrals. Be-
sides classical material (Calder6n-Zygmund and Littlewood-Paley theory),
this article eontains an exposition of reeent results, whieh in an essential
way have widened the seope of the whole area and have made it possible to
solve many old problems, thereby sometimes transeending the very frames of
harmonie analysis in its eanonieal interpretation.
Quite different but highly interesting and often tantalizing material is eol-
leeted in S. V. Kislyakov's eontribution, whieh eoncludes this volume, cen-
tering around the not ion of "exeeptional" (or "narrow") sets (this topie was
brießy diseussed already in V. P. Khavin's eontribution in the first volume of
this series). Special attention is given to the so-ealled Sidon set, whieh are
eonnected with some important results obtained in recent years, and further
to methods utilizing the not ion of eapacity, whieh has its origin in potential
theory.
V. P. Khavin, N. K. Nikol'skil
List of Editors, Authors and Translators
Editor-in-Chief
R. V. Gamkrelidze, Academy of Sciences of the USSR, Steklov Mathematical
Institute, ul. Vavilova 42, 117966 Moscow, Institute for Scientific Informa-
tion (VINITI), ul. Usievicha 20a, 125219 Moscow, USSR
Consulting Editors
V. P. Khavin, Department of Mathematics, Leningrad University, Staryi Pe-
terhof, 198094 Leningrad, USSR
N. K. Nikol'ski'l, Steklov Mathematical Institute, Fontanka 27, 191011 Lenin-
grad, USSR
Authors
Sh. A. Alimov, Tashkent State University, Vuzgorodok, 700095 Tashkent,
USSR
R. R. Ashurov, Tashkent State University, Vuzgorodok, 700095 Tashkent,
USSR
E. M. Dyn'kin, Department of Mathematics, Leningrad Institute of Electrical
Engineering, Prof. Popov st. 5, Leningrad, USSR
S. V. Kislyakov, Steklov Mathematical Institute, Fontanka 27, 191011 Lenin-
grad, USSR
A. K. Pulatov, Tashkent State University, Vuzgorodok, 700095 Tashkent,
USSR
Translator
J. Peetre, Matematiska institutionen, Stockholms universitet, Box 6701,
S-113 85 Stockholm, Sweden
Contents
I. Multiple Fourier Series and Fourier Integrals
Sh. A. Alimov, R. R. Ashurov, A. K. Pulatov
1
11. Methods of the Theory of Singular Integrals:
Littlewood-Paley Theory and Its Applieations
E. M. Dyn'kin
97
111. Exeeptional Sets in Harmonie Analysis
S. V. Kislyakov
195
Author Index
223
Subject Index
226
I. Multiple Fourier Series and
Fourier Integrals
Sh. A. Alimov, R. R. Ashurov, A. K. Pulatov
Translated from the Russian
by J. Peetre
Contents
Introduetion . . . . . . . . . . . . . . . . . . . 3
1. How Multiple Trigonometrie Series Arise . . . . 3
2. What Do We Mean by the Sum of the Series (I)? 6
3. Various Partial Sums of a Multiple Series 7
4. Forms of Convergenee . . 9
5. Eigenfunction Expansions . 10
6. Summation Methods . . . 13
7. Multiple Fourier Integrals . 15
8. The Kernel of the Fourier Integral Expansion 17
9. The Diriehlet Kernel . . . . . . . . . . . 20
10. Classes of Differentiable Functions . . . . 23
11. A Few Words About Further Developments 27
Chapter 1. Loealization and Uniform Convergenee 27
§1. The Loealization Principle . . . . . . . . 27
1.1. On the Loealization Problem . . . . . 27
1.2. Loealization of Reetangular Partial Sums 28
1.3. Loealization of Spherieal Partial Integrals 31
1.4. Equieonvergenee of the Fourier Series and the Fourier Integral 32
1.5. Riesz Means Below the Critical Index. . . . . . . . . 34
1.6. Loealization Under Summation over Domains Whieh Are
Level Sets of an Elliptic Polynomial 36
§2. Uniform Convergenee. . . . . 37
2.1. The One-Dimensional Case 37
2.2. Reetangular Sums . . . . 38
2 Sh. A. Alimov, R. R. Ashurov, A. K. Pulatov
2.3. Uniform Convergence of Spherical Sums .... 40
2.4. Summation over Domains Bounded by the Level Surfaces
of an Elliptic Polynomial . . . 41
Chapter 2. Lp - Theory . . . . . . . . 42
§1. Convergence of Fourier Series in Lp 42
1.1. Rectangular Convergence . . . 42
1.2. Circular Convergence . . . . . 43
1.3. Summation of Spherical lliesz Means 45
§2. Convergence of Fourier Integral Expansions in Lp 49
2.1. The Case N = 1 . . . . . . . . . . . . . 49
2.2. Bases and the Problem of Spherical Multipliers 50
2.3. Bases of lliesz Means . . . . . 52
§3. Multipliers . . . . . . . . . . . 54
3.1. The Theorem of Marcinkiewicz 54
3.2. The Nonperiodic Case . . . . 56
3.3. Are Multipliers in RN Multipliers on TN ? 58
3.4. Pseudo-Differential Operators . . . 60
3.5. Fourier Integral Operators .... 65
Chapter 3. Convergence Almost Everywhere 67
§1. Rectangular Convergence . . 67
1.1. Quadratic Convergence . . . . . . 67
1.2. Rectangular Convergence . . . . . 69
§2. Convergence a. e. of Spherical Sums and Their Means 72
2.1. Convergence of Spherical Sums .... 72
2.2. Convergence a. e. of Sperical Riesz Means 73
Chapter 4. Fourier Coefficients . . 78
§1. The Cantor-Lebesgue Theorem 78
1.1. The One-Dimensional Case 78
1.2. Spherical Partial Sums 78
§2. The Denjoy-Luzin Theorem 80
2.1. The Case N = 1 . . 80
2.2. The Spherical Mean . 81
2.3. Rectangular Sums . . 82
§3. Absolute Convergence of Fourier Series 83
3.1. Some One-Dimensional Results 83
3.2. Absolute Convergence of Multiple Series 84
3.3. On the Convergence of the Series of Powers of Ilnl 87
Remarks and Bibliographical Notes 87
Bibliography . . . . . . . . . . . . . . . . . . . . 91
I. Multiple Fourier Series and Fourier Integrals 3
Introdu ction
1. How Multiple Trigonometrie Series Arise. The basie objects of study
in this article are functions of several variables whieh are periodic in each of
these variables. We may assume (this assumption is not very restrietive) that
the eorresponding periods are the same and equal to 211". Thus, we may take
the fundamental set, where our functions are defined, to be the N-dimensional
eube
1l'N = {x E RN: -11" < Xj ~ 1I",j = 1, ... ,N}.
This eube ean be identified in a natural way with an N -dimensional torus
(i. e. the subset of CN of the form (eiXl, • .• , eiXN ), where (Xl, . .. , XN) ERN),
whieh in what follows we shall not distinguish from 1l'N.
The fundamental harmonies with respect to whieh we take our expansion
have in appearanee exactly the same form as in the one-dimensional ease:
einx where, for N > 1, nx stands for the inner produet
nx = nlXl + ... + nNXN , nE ZN, x E RN.
Here, as usual, ZN is the set of all vectors with integer eomponents: n =
(nl, ... ,nN) is in ZN ifnj = 0,±1, ....
A multiple trigonometrie series
(1)
where the eoefficients Cn are arbitrary eomplex numbers, looks exactly as
an ordinary "one-dimensional" series. This exterior resemblanee leads one in
a eompletely natural way to the idea that the multidimensional version of
Fourier analysis is subject to the general principles whieh, in partieular, were
diseussed in the introduetory article of the series "Commutative Harmonie
Analysis" (Khavin (1988)). We run into a similar situation when we eonsider
the vibrations of astring, whieh is subjeet to the same principles as the
vibrations of eomplieated spatial objeets. This analogy, whieh may appear
superficial, has in fact a deep interior meaning.
Multidimensional harmonie analysis largely owes its appearanee to the
study of various vibrating systems. As is weIl-lmown, the early 19th eentury
witnessed new progress in the mathematieal deseription of physieal proeesses
taking place in real space, without assumptions of symmetry, whieh often
reduee the situation to functions of one variable. In the first place let us
here mention the classieal work of Fourier in the theory of the distribution
of heat (1807, the detailed study appearing in 1822), as weIl as the work of
Laplace and Diriehlet, where not only the eelebrated "method of Fourier"
was set forth but, essentially, also a eonsiderable portion of the problems of
harmonie analysis were formulated whieh eame to determine the development
of this subjeet for a long period of years.