Approximation by Multivariate Singular Integrals
Approximation by Multivariate Singular Integrals is the first monograph to illustrate the approximation of multivariate singular integrals to the identity-unit operator. The basic approximation properties of the general multivariate singular integral oper
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George A. Anastassiou
Approximation by Multivariate Singular Integrals
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George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152 USA [email protected]
ISSN 2191-8198 e-ISSN 2191-8201 ISBN 978-1-4614-0588-7 e-ISBN 978-1-4614-0589-4 DOI 10.1007/978-1-4614-0589-4 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011933230 Mathematics Subject Classification (2010): 41A17, 41A25, 41A28, 41A35, 41A36, 41A60, 41A80 © George A. Anastassiou 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To my wife Koula and my daughters Angela and Peggy
Preface
This short monograph is the first to deal exclusively with the study of the approximation of multivariate singular integrals to the identity-unit operator. Here we study quantitatively the basic approximation properties of the general multivariate singular integral operators, special cases of which are the multivariate Picard, Gauss-Weierstrass, Poisson-Cauchy and trigonometric singular integral operators, etc. These operators are not general positive linear operators. In particular we study the rate of convergence of these operators to the unit operator, as well as the related simultaneous approximation. These are given via inequalities and by the use of multivariate higher order modulus of smoothness of the high order partial derivatives of the involved function. Also we study the global smoothness preservation properties of these operators. Some of these multivariate inequalities are proved to be attained, that is sharp. Furthermore we give asymptotic expansions of Voronovskaya type for the error of approximation. These properties are studied with respect to Lp norm, 1 p 1: The last chapter presents a related Korovkin type approximation theorem for functions of two variables. Plenty of examples are given. For the convenience of the reader, the chapters are self-contained. This brief monograph relies on author’s last two years of related research work, more precisely see author’s articles in the list of references of each chapter. Advanced courses can be taught out of this short book. All necessary background and motivations are given per chapter. The presente
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