Abstract Harmonic Analysis of Continuous Wavelet Transforms
This volume contains a systematic discussion of wavelet-type inversion formulae based on group representations, and their close connection to the Plancherel formula for locally compact groups. The connection is demonstrated by the discussion of a toy exam
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Hartmut F¨uhr
Abstract Harmonic Analysis of Continuous Wavelet Transforms
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Author Hartmut F¨uhr Institute of Biomathematics and Biometry GSF - National Research Center for Environment and Health Ingolst¨adter Landstrasse 1 85764 Neuherberg Germany e-mail: [email protected]
Library of Congress Control Number: 2004117184
Mathematics Subject Classification (2000): 43A30; 42C40; 43A80 ISSN 0075-8434 ISBN 3-540-24259-7 Springer Berlin Heidelberg New York DOI: 10.1007/b104912 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science + Business Media http://www.springeronline.com c Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors 41/3142/du - 543210 - Printed on acid-free paper
Preface
This volume discusses a construction situated at the intersection of two different mathematical fields: Abstract harmonic analysis, understood as the theory of group representations and their decomposition into irreducibles on the one hand, and wavelet (and related) transforms on the other. In a sense the volume reexamines one of the roots of wavelet analysis: The paper [60] by Grossmann, Morlet and Paul may be considered as one of the initial sources of wavelet theory, yet it deals with a unitary representation of the affine group, citing results on discrete series representations of nonunimodular groups due to Duflo and Moore. It was also observed in [60] that the discrete series setting provided a unified approach to wavelet as well as other related transforms, such as the windowed Fourier transform. We consider generalizations of these transforms, based on a representationtheoretic construction. The construction of continuous and discrete wavelet transforms, and their many relatives which have been studied in the past twenty years, involves the following steps: Pick a suitable basic element (the wavelet) in a Hilbert space, and construct a system of vectors from it by the action of certain prescribed operators on the basic element, with the aim of expanding arbitrary elements of the Hilbert space in this system. The associated wavelet transform is the map which assigns each element of the Hilbert space its expansion coefficients, i.e. the family of
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