Current Topics in Summability Theory and Applications
This book discusses recent developments in and contemporary research on summability theory, including general summability methods, direct theorems on summability, absolute and strong summability, special methods of summability, functional analytic methods
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rent Topics in Summability Theory and Applications
Current Topics in Summability Theory and Applications
Hemen Dutta Billy E. Rhoades •
Editors
Current Topics in Summability Theory and Applications
123
Editors Hemen Dutta Department of Mathematics Gauhati University Guwahati, Assam India
ISBN 978-981-10-0912-9 DOI 10.1007/978-981-10-0913-6
Billy E. Rhoades Department of Mathematics Indiana University Bloomington, IN USA
ISBN 978-981-10-0913-6
(eBook)
Library of Congress Control Number: 2016931604 © Springer Science+Business Media Singapore 2016 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. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Science+Business Media Singapore Pte Ltd.
Preface
This book is intended for graduate students and researchers who have interest in functional analysis, in general and summability theory, in particular. It describes several useful topics in summability theory along with applications. The book consists of nine chapters and is organized as follows: Chapter “An Introduction to Summability Methods” is introductory in nature. This chapter focuses on the historical development of summability theory right from Cauchy’s concept to till date. Summability methods developed from the two basic processes—T-process and /-process—have also been discussed in this chapter. Chapter “Some Topics in Summability Theory” deals with the study of some classical and modern summability methods, and the connections among them. In fact, results concerning summability by weighted mean method, the ðM; kn Þ method, the Abel method, and the Euler method are presented. Then the sequence space Kr , r 1 being a fixed integer, is defined and a Steinhaus type theorem is proved. The space Kr is then studied in the context of sequences of 0’s and 1’s. Further, the core of a sequence is studied, an improvement of a result of Sherbakhoff is proved and a very simple proof of Knopp’s core theorem is then
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