Fractal-Based Methods in Analysis
The idea of modeling the behavior of phenomena at multiple scales has become a useful tool in both pure and applied mathematics. Fractal-based techniques lie at the heart of this area, as fractals are inherently multiscale objects; they very often describ
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Herb Kunze • Davide La Torre • Franklin Mendivil Edward R. Vrscay
Fractal-Based Methods in Analysis
Herb Kunze Department of Mathematics and Statistics University of Guelph Guelph Ontario Canada [email protected]
Davide La Torre Department of Economics, Business and Statistics University of Milan Milan Italy [email protected]
Franklin Mendivil Department of Mathematics and Statistics Acadia University Wolfville Nova Scotia Canada [email protected]
Edward R. Vrscay Department of Applied Mathematics University of Waterloo Waterloo Ontario Canada [email protected]
e-ISBN 978-1-4614-1891-7 ISBN 978-1-4614-1890-0 DOI 10.1007/978-1-4614-1891-7 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011941791 Mathematics Subject Classification (2010): 28A80 , 28A33, 1AXX, 45Q05, 28CXX, 65L09, 65N21 © Springer Science+Business Media, LLC 2012 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)
In memory of Bruno Forte mentor, teacher, collaborator, friend.
Preface
The idea of modeling the behaviour of phenomena at multiple scales has become a useful tool in both pure and applied mathematics. Fractal-based techniques lie at the heart of this area, as fractals are inherently multiscale objects. Fractals have increasingly become a useful tool in real applications; they very often describe such phenomena better than traditional mathematical models. Fractal-Based Methods in Analysis draws together, for the first time in book form, methods and results from almost 20 years of research on this topic, including new viewpoints and results in many of the chapters. For each topic, the theoretical framework is carefully explained. Numerous examples and applications are presented. The central themes are self-similarity across scales (exact or approximate) and contractivity. In applications, this involves introducing an appropriate space for contractive operators and approximating the “target” mathematical object by the fixed point of one of these contractions. Under fairly general conditions, this approximation can be extremely good. This idea emerged from fractal image compression, where an image is encoded by the parameters of a contractive transformation (see Sect. 3.1 and Figs. 3.3 and 3.4). The first step in exten
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