Approximation by Max-Product Type Operators
This monograph presents a broad treatment of developments in an area of constructive approximation involving the so-called "max-product" type operators. The exposition highlights the max-product operators as those which allow one to obtain, in many cases,
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oximation by Max-Product Type Operators
Approximation by Max-Product Type Operators
Barnabás Bede • Lucian Coroianu • Sorin G. Gal
Approximation by Max-Product Type Operators
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Barnabás Bede Department of Mathematics DigiPen Institute of Technology Redmond, WA, USA
Lucian Coroianu Department of Mathematics and Computer Science University of Oradea Oradea, Romania
Sorin G. Gal Department of Mathematics and Computer Science University of Oradea Oradea, Romania
ISBN 978-3-319-34188-0 ISBN 978-3-319-34189-7 (eBook) DOI 10.1007/978-3-319-34189-7 Library of Congress Control Number: 2016940388 Mathematics Subject Classification (2010): 41A35, 41A20, 41A25, 41A27, 41A40, 41A29, 41A30, 41A05, 94A12, 47H10, 28A80 © Springer International Publishing Switzerland 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 International Publishing AG Switzerland
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Preface
In this research monograph, we bring to light an interesting new direction in constructive approximation of functions by operators. There are at least two natural justifications for these new operators, termed by us as max-product operators (for reasons we will see below). They are based on possibility theory, a mathematical theory dealing with certain types of uncertainties and which is considered as an alternative to probability theory. The first justification is based on the interpretations of the max-product Bernstein operator as a possibilistic expectation of a particular fuzzy variable having a possibilistic Bernoulli distribution and on a Chebyshev-type inequality in the theory of possibility, facts which are in a perfect analogy with the probabilistic approach of Bernstein for the convergence of the classical Bernstein polynomials. The second justification is based on the Feller scheme in terms of the possibilistic integral, which again is in perfect analogy w
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