Approximation with Positive Linear Operators and Linear Combinations
This book presents a systematic overview of approximation by linear combinations of positive linear operators, a useful tool used to increase the order of approximation. Fundamental and recent results from the past decade are described with their correspo
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Vijay Gupta Gancho Tachev
Approximation with Positive Linear Operators and Linear Combinations
Developments in Mathematics Volume 50
Series editors Krishnaswami Alladi, Gainesville, USA Hershel M. Farkas, Jerusalem, Israel
More information about this series at http://www.springer.com/series/5834
Vijay Gupta • Gancho Tachev
Approximation with Positive Linear Operators and Linear Combinations
123
Vijay Gupta Department of Mathematics Netaji Subhas Institute of Technology New Delhi, India
Gancho Tachev Department of Mathematics University of Architecture Civil Engineering and Geodesy Sofia, Bulgaria
ISSN 1389-2177 ISSN 2197-795X (electronic) Developments in Mathematics ISBN 978-3-319-58794-3 ISBN 978-3-319-58795-0 (eBook) DOI 10.1007/978-3-319-58795-0 Library of Congress Control Number: 2017940878 Mathematics Subject Classification: 41A25, 41A30, 30E05, 30E10 © Springer International Publishing AG 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 International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface
Approximation of functions by positive linear operators is an important branch of the approximation theory. To increase the order of approximation a useful tool is the method of linear combinations of positive linear operators (p.l.o.). The most known example of p.l.o. is the famous Bernstein operators introduced by S. Bernstein [26, 27], which for f 2 CŒ0; 1 is given by Bn .f ; x/ D
n X kD0
pn;k .x/f .k=n/;
! n k pn;k .x/ D x .1 x/nk : k
In 1932 Elena Voronovskaja [193]—a doctoral student of S. Bernstein—proved that if f is bounded on Œ0; 1, differentiable in some neighbourhood of x and has second derivative f 00 for some x 2 Œ0; 1 then lim nŒBn .f ; x/ f .x/ D
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