Absolute Summability of Fourier Series and Orthogonal Series
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1067 Yasuo Okuyama
Absolute Summability of Fourier Series and Orthogonal Series
Springer-Verlag Berlin Heidelberg New York Tokyo 1984
Author
Yasuo Okuyama Department of Mathematics, Faculty of Engineering Shinshu University Wakasato, Nagano 380, Japan
AMS Subject Classification (1980): 42A28, 42C15
ISBN 3-540-13355-0 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13355-0 Springer-Verlag New York Heidelberg Berlin Tokyo Library of Congress Cataloging in Publication Data. Okuyama, Yasuo, 1937- Absolute summability of Fourier series and orthogonal series. (Lecture notes in mathematics; 1067) Bibliography: p. Includes index. 1. Fourier series. 2. Series, Orthogonal. 3. Summability theory. I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag); 1067. QA3.L28 no. 1067 [QA404] 510s [515'.2433] 84-10713 ISBN 0-387-13355-0 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
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Preface The purpose of these lecture notes is to study the absolute summability of Fourier series and orthogonal series. The absolute summability is a generalization of the concept of the absolute convergence just as the summability is an extension of the concept of the convergence. The absolute convergence of Fourier series is closely related with the variation and the modulus of continuity of functions. There are several classical criteria for the absolute convergence of Fourier series. We would like to show that these criteria can be systematically proved from the point of view of the best approximation and that we can offer some applications. On the other hand, we consider the absolute summabilitieslN,Pnl and IR,Pn,ll for the nonabsolute convergent Fourier series and orthogonal series. Then we can extend the concepts of the absolute convergence of Fourier series and orthogonal series by the absolute summability just as we can do to the convergence of Fourier series by the summability. Consequently we can give several criteria for the nonabsolute convergent Fourier series and orthogonal series systematically. The obJect of Chapter 1 is to make clear the background of the absolute convergence of Fourier series for both the trigonometrical system and the Walsh system with the aid of Stechkin's Theorem and an inequality on the best approximation and give also the similar results for orthogonal systems satisfying some conditions. In Chapter 2 we deal with the absolute Norlund summability almost everywhere of Fourier series and, from Theorem 2.9 which is equivalent to Lal's Theorem [38,39J, we deduce
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