Affine Density in Wavelet Analysis
In wavelet analysis, irregular wavelet frames have recently come to the forefront of current research due to questions concerning the robustness and stability of wavelet algorithms. A major difficulty in the study of these systems is the highly sensitive
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Gitta Kutyniok
Affine Density in Wavelet Analysis
1914
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1914
Gitta Kutyniok
Affine Density in Wavelet Analysis
ABC
Author Gitta Kutyniok Until September 2007 Program in Applied and Computational Mathematics Princeton University Princeton, NJ 08544 USA e-mail: [email protected] From October 2007 Department of Statistics Stanford University Stanford, CA 94305 USA e-mail: [email protected]
Library of Congress Control Number: 2007928330 Mathematics Subject Classification (2000): 42C40, 42C15, 94A12 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN 978-3-540-72916-7 Springer Berlin Heidelberg New York DOI 10.1007/978-3-540-72949-5 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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 springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, 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. Typesetting by the author and SPi using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper
SPIN: 12070794
41/SPi
543210
Dedicated to
my Parents
Preface
During the last 20 years, wavelet analysis has become a major research area in mathematics, not only because of the beauty of the mathematical theory of wavelet systems (sometimes also called affine systems), but also because of its significant impact on applications, especially in signal and image processing. After the extensive exploration of orthonormal bases of classical affine systems that has occupied much of the history of wavelet theory, recently both wavelet frames — redundant wavelet systems — and irregular wavelet systems — wavelet systems with an arbitrary sequence of time-scale indices — have come into focus as a main area of research. Two main reasons for this are to serve new applications which require robustness against noise and erasures, and to derive a deeper understanding of the theory of classical affine systems. However, a comprehensive theory to treat irregular wavelet frames does not exist so far. The main difficulty consists of the highly sensitive interplay between geometric properties of the sequence of time-scale indices and frame properties of the associated wavelet sy
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