Trigonometric Fourier Series and Their Conjugates

Trigonometric Fourier Series and Their Conjugates

Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science. Amsterdam. The Netherlands Volume 372

Trigonometric Fourier Series and Their Conjugates by Levan Zhizhiashvili Department a/mechanics and Mathematics, Tbilisi State University, Tbilisi, Georgia KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON

A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN-13: 978-94-010-6612-9 DOl: 10.1007/978-94-009-0283-1 e-isbn-13: 978-94-009-0283-1 Published by Kluwer Academic Publishers, P.O. Box 17,3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands. This is a revised and updated translation of the original Russian work Some Problems of the Theory of Trigonometric Fourier Series and Their Conjugate Series, Tbilisi State University Press, Tbilisi, Georgia 1993. Translated by George K vinikadze. All Rights Reserved 1996 Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

In fond memory of V. G. CHELIDZE AND D. E. MEN'SHOV

Table of Contents Preface... Xl Part 1 SIMPLE TRIGONOMETRIC SERIES Chapter I. The Conjugation Operator and the Hilbert Transform 1.1. Some Definitions and Well-known Results... 3 1.2. On the Theorems of Kolmogorov, Titchmarsh and Kober... 15 1.3. On a Theorem of Hardy and Littlewood... 27 1.4. Unsolved Problems.................................................. 33 Chapter II. Pointwise Convergence and Summability of Trigonometric Series 2.1. Some Definitions and Well-knuwn Results... 35 2.2. Pointwise Convergence and Summability of Fourier Series and Conjugate Trigonometric Series... 43 2.3. On the Divergence Almost Everywhere of Trigonometric Fourier Series... 61 2.4. Unsolved Problems... 68 Chapter III. Convergence and Summability of TrigonOl!letri.: Fourier Series and Their Conjugates in the Spaces LP(T), p E JO, +oo[ 3.1. Convergence and Summability of the Series O"[JJ and a[fj in the Spaces LP(T), p E]O, 1]... 71 3.2. Convergence and Summability of O"[fJ and a[fj in the Spaces LP(T), p E [1, +ooj... 72 3.3. On the Series O"[fJ and O"[lJ for Even and Odd f... 85 3.4. Unsolved Problems... 92 Chapter IV. Some Approximating Properties of Cesaro Means of the Series O"[fJ and a[jj 4.1. Approximating Properties of Cesaro Means of O"[JJ... 93 4.2. Approximating Properties of Cesaro Means of the Series a[lj... 109 4.3. Unsolved Problems... 116

viii Part 2 MULTIPLE TRIGONOMETRIC SERIES Chapter I. Conjugate Functions and Hilbert Transforms of Functions of Several Variables 1.1. Some Definitions and Auxiliary Statements... 119 1.2. On Existence of Conjugate Functions and of Hilbert Transforms... 132 1.3. Lebesgue Integrability of Conjugate Functions of Several Variables... 152 1.4. On the Validity of a Theorem of M. Riesz for Functions of Several Variables... 156 1.5. On the Validity of Kolmogorov's Theorem for Conjugate Functions and Hilbert Transforms of Functions of Several Variables. '" "....... 161 1.6. On Some Conditions of Existence and Integrability of Conjugate Functions of Several Variables in Terms of Mixed and Partial Integral Moduli of Continuity... 164 1.7. Unsolved Problems... 166 Chapter II. Convergence and Summability at a Point or Almost Everywhere of Multiple Trigonometric Fourier Series and Their Conjugates 2.1. Convergence and Summability by Multiple Cesaro Method of Negative Order of the Series 17n!!] and an!!, B] at Separate Points or Almost Everywhere................................... 167 2.2. Almost Everywhere Summability of the Series an!!, B] by Multiple Methods of Poisson-Abel and Cesaro of Positive Order. Integrability of Some Majorants Connected with Multiple Series 17n [f] and an[f, B]... 182 2.3. Unsolved Problems... ". "... 201 Chapter III. Some Approximating Properties of n-fold Cesaro Means ofthe Series 17n!!] and an[f,b] 3.1. Approximating Properties of n-fold Cesaro Means of the Series 17n!!]... 205 3.2. Approximating Properties of n-fold Cesaro Means of the Series an!!, B]... 223 3.3. Unsolved Problems... 224 Chapter IV. Convergence and Summability of Multiple Trigonometric Fourier Series and Their Conjugates in the Spaces LP(Tn), p E ]0, +00] 4.1. Convergence and Summability of the Series O"n[f] and an!!, B] in the Spaces LP(Tn), p E ]0,1]... 225 4.2. Convergence and Summability of the Series 17n [f] and an[j, B] in the Spaces LP(Tn), p E [1, +00]... 230 4.3. Unsolved Problems... 241

Chapter V. Summability of Series CT2[J] and (72[f, B] by a Method of the Marcinkiewicz Type 5.1. Summability Almost Everywhere of Series CT2[f] and (72[J, B] by a Method of the Marcinkiewicz Type... 243 5.2. Some Approximating Properties of Marcinkiewicz Type Means of the Series CT2[f] in the Spaces LP(T2), p E [1, +00]... 255 5.3. Unsolved Problems... 266 Bib I i 0 g rap h y.............. 267 In d ex... 299 IX

Preface Research in the theory of trigonometric series has been carried out for over two centuries. The results obtained have greatly influenced various fields of mathematics, mechanics, and physics. Nowadays, the theory of simple trigonometric series has been developed fully enough (we will only mention the monographs by Zygmund [15, 16] and Bari [2]). The achievements in the theory of multiple trigonometric series look rather modest as compared to those in the one-dimensional case though multiple trigonometric series seem to be a natural, interesting and promising object of investigation. We should say, however, that the past few decades have seen a more intensive development of the theory in this field. To form an idea about the theory of multiple trigonometric series, the reader can refer to the surveys by Shapiro [1], Zhizhiashvili [16], [46], Golubov [1], D'yachenko [3]. As to monographs on this topic, only that ofyanushauskas [1] is known to me. This book covers several aspects of the theory of multiple trigonometric Fourier series: the existence and properties of the conjugates and Hilbert transforms of integrable functions; convergence (pointwise and in the LP-norm, p > 0) of Fourier series and their conjugates, as well as their summability by the Cesaro (C,a), a> -1, and Abel-Poisson methods; approximating properties of Cesaro means of Fourier series and their conjugates. The structure of the book is motivated by my wish to place a special emphasis on the new effects which arise while dealing with multiple series and which are not inherent in the one-dimensional case. In Part 1, the familiar results concerning functions of one variable are discussed, while in Part 2, the material is parallelled for the multidimensional case. There the results obtained by the author are presented, with many proofs being published for the first time. At the beginning of each chapter, a brief survey of known results is given, and at the end, unsolved problems are formulated whose solution, in my opinion, will greatly contribute to the development of the theory. I hope the book will prove useful both to research workers and graduate students whose scientific interests lie in this field. It should be noted that the book is not self-contained and assumes the reader has sufficient background knowledge. The Russian original of this book was first published in 1993 in Tbilisi (Georgia), but it has not really become widely available to the majority of potential readers. So I am very grateful to Kluwer Academic Publishers for their interest in the English translation of the book. The English version contains some minor changes and improvements. I wish to express my sincere appreciation to Prof. P. L. Vl'yanov, corresponding member of the Russian Academy of Sciences, Prof. B. I. Golubov and Prof. V. Kokilashvili who read the manuscript and offered valuable advice.

xii I am also grateful to my colleagues from Tbilisi State University A. Ambroladze, T. Akhobadze, G. Bareladze, V. Bugadze, R. Getsadze, M. Lekishvili, D. Leladze, L. Panjikidze, T. Tevzadze, G. Tkebuchava for help during the work on the manuscript and proofreading both of the original and of the translation. I am especially thankful to my colleagues G. Kvinikadze and M. Kvinikadze, the former for translating the book and the latter for preparing the camera-ready typescript. Tbilisi, November 1995 LEVAN ZmzHIASHVlLI