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   -> Volume 11, Issue 4

Book: Wavelets through a Looking Glass: The World of the Spectrum, by Ola Bratteli and Palle Jorgensen
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Brian Treadway (

PostPosted: Sat Mar 15, 2003 1:12 am    
Subject: Book: Wavelets through a Looking Glass: The World of the Spectrum, by Ola Bratteli and Palle Jorgensen
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Summary: Recent developments in wavelet theory; fundamental, timeless techniques with geometric, spectral-theoretic flavor. Clearly motivated, systematic exposition. Excellent graphics. New results: homotopy of multiresolutions, spectrum of transfer operators

Ola Bratteli, University of Oslo, Norway
Palle Jorgensen, University of Iowa, Iowa City, IA

This book combining wavelets and the world of the spectrum focuses on recent developments in wavelet theory, emphasizing fundamental and relatively timeless techniques that have a geometric and spectral-theoretic flavor. The exposition is clearly motivated and unfolds systematically, aided by numerous graphics.

Key features of the book:
* The important role of the spectrum of a transfer operator is studied
* Excellent graphics show how wavelets depend on the spectra of the transfer operators
* Key topics of wavelet theory are examined: connected components in the variety of wavelets, the geometry of winding numbers, the Galerkin projection method, classical functions of Weierstrass and Hurwitz and their role in describing the eigenvalue-spectrum of the transfer operator, isospectral families of wavelets, spectral radius formulas for the transfer operator, Perron-Frobenius theory, and quadrature mirror filters
* New previously unpublished results appear on the homotopy of multiresolutions, on approximation theory, and on the spectrum and structure of the fixed points of the associated transfer and subdivision operators
*Concise background material for each chapter, open problems, exercises, bibliography, and comprehensive index make this work a fine pedagogical and reference resource.
* This self-contained book deals with the tools for important applications to signal processing, communications engineering, computer graphics algorithms, qubit algorithms and chaos theory, and is aimed at a broad readership of graduate students, practitioners, and researchers in applied mathematics and engineering. The book is also useful for other mathematicians with an interest in the interface between mathematics and communication theory.

Publisher's web site, with ordering information:

Authors' web sites, with full Table of Contents and other excerpts:

July 2002/xxii+398 pp., 147 illus./Hardcover/$59.95/ISBN 0-8176-4280-3
Applied and Numerical Harmonic Analysis Series

Reader's review of Wavelets Through A Looking Glass: The World of the Spectrum, by O. Bratteli and P. Jorgensen

This book does a superlative job of demonstrating the richness of the theory of wavelets, which began as an outgrowth of classical harmonic analysis. The authors have demonstrated further connections with spectral theory, ergodic theory, homotopy theory and the theory of probability---just to name a few of the well-established areas of mathematics which are shown to touch the theory of wavelets. At the same time the material is beautifully documented by means of 61 figures, numerous tables and other illustrations which are freely distributed throughout the book. An exhaustive list of 200 references to the most current literature ensures scholarly care and the most up-to-date account of the topics covered.

The authors succeed admirably in achieving the two-fold purpose of the book. On the one hand the goal is to give a modern (but "timeless") presentation of wavelet theory while on the other hand the goal is to present new results that have not previously been published. The latter include material on homotopy of resolutions, approximation theory and results on the spectrum of the associated transfer operators and subdivision operators. The first goal is well-served by the many well-documented exercises which appear at the end of each chapter. The pedagogy is further enhanced by several paragraphs of illuminating prose at the beginning of each chapter---to set the stage for the technical material to follow. Although not written as a conventional text, one would expect that an industrious graduate student could profit enormously from a serious exposure to this book.

With respect to the literature on wavelets, it is difficult to recall any other book that is so well documented both with graphical and numerical details as well as mathematical proof. This volume will remain a central reference work for many years to come.

Mark Pinsky,
Northwestern University

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