Core of a Sequence and the Matrix Class \((\ell , \ell )\)
In this chapter, we define the core of a sequence and prove an improvement of Sherbakhoff’s result, which gives rise to a short and elegant proof of Knopp’s core theorem . We also present some nice properties of the class \((\ell , \ell )\) of infinite ma
- PDF / 1,430,202 Bytes
- 139 Pages / 453.543 x 683.15 pts Page_size
- 18 Downloads / 198 Views
Classical Summability Theory
Classical Summability Theory
P.N. Natarajan
Classical Summability Theory
123
P.N. Natarajan Formerly of the Department of Mathematics Ramakrishna Mission Vivekananda College Chennai, Tamil Nadu India
ISBN 978-981-10-4204-1 DOI 10.1007/978-981-10-4205-8
ISBN 978-981-10-4205-8
(eBook)
Library of Congress Control Number: 2017935546 © Springer Nature Singapore Pte Ltd. 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore
Dedicated to my wife Vijayalakshmi and my children Sowmya and Balasubramanian
Preface
The study of convergence of infinite series is a very old art. In ancient times, people were interested in orthodox examination for convergence of infinite series. Divergent series, i.e., infinite series which do not converge, was of no interest to them until the advent of L. Euler (1707–1783), who took up a serious study of divergent series. He was later followed by a galaxy of very great mathematicians. Study of divergent series is the foundation of summability theory. Summability theory has many utilities in analysis and applied mathematics. An engineer or physicist, who works on Fourier series, Fourier transforms or analytic continuation, can find summability theory very useful for his/her research. In the present book, some of the contributions of the author to classical summability theory are highlighted, thereby supplementing, the material already available in standard texts on summability theory. There are six chapters in all. The salient features of each chapter are listed below. In Chap. 1, after a very brief introduction, we recall well-known definitions and concepts. We stat
