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Wavelets, Vibrations and Scalings
Yves Meyer, University of Paris IX, France
A co-publication of the AMS and Centre de Recherches Mathématiques.
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CRM Monograph Series
1998; 133 pp; hardcover
Volume: 9
ISBN-10: 0-8218-0685-8
ISBN-13: 978-0-8218-0685-2
List Price: US$36
Member Price: US$28.80
Order Code: CRMM/9
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Physicists and mathematicians are intensely studying fractal sets of fractal curves. Mandelbrot advocated modeling of real-life signals by fractal or multifractal functions. One example is fractional Brownian motion, where large-scale behavior is related to a corresponding infrared divergence. Self-similarities and scaling laws play a key role in this new area.

There is a widely accepted belief that wavelet analysis should provide the best available tool to unveil such scaling laws. And orthonormal wavelet bases are the only existing bases which are structurally invariant through dyadic dilations.

This book discusses the relevance of wavelet analysis to problems in which self-similarities are important. Among the conclusions drawn are the following: 1) A weak form of self-similarity can be given a simple characterization through size estimates on wavelet coefficients, and 2) Wavelet bases can be tuned in order to provide a sharper characterization of this self-similarity.

A pioneer of the wavelet "saga", Meyer gives new and as yet unpublished results throughout the book. It is recommended to scientists wishing to apply wavelet analysis to multifractal signal processing.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

Readership

Graduate students, research mathematicians, physicists, and other scientists working in wavelet analysis.

Reviews

"This monograph grew out of five lectures given by the author at the University of Montreal on the theme of multifractal analysis. It exposes and completes the work of S. Jafffard and the author on pointwise regularity and local oscillations of functions, and several mathematical aspects of the recent work of A. Arnéodo on multifractals are studied."

-- Mathematical Reviews

"Meyer's book is sprinkled throughout with a fascinating collection of examples and counter-examples. Meyer's book is suitable for professional researchers in function theory, or as a text for an advanced graduate seminar. Meyer writes in a compressed yet lucid style which invites the reader's participation."

-- Bulletin of the London Mathematical Society

"Tools from applications have been used in wavelet analysis to great advantage, and powerful methods from wavelet algorithms have in turn found an impressive host of recent practical applications. This exchange of ideas is masterfully brought to light in Meyer's book. Meyer giving lucid explanation of the key concepts."

-- Palle Jorgensen

Table of Contents

  • Introduction
  • Scaling exponents at small scales
  • Infrared divergences and Hadamard's finite parts
  • The 2-microlocal spaces \(C^{s,s^{\prime}}_{x_0}\)
  • New characterizations of the two-microlocal spaces
  • An adapted wavelet basis
  • Combining a Wilson basis with a wavelet basis
  • Bibliography
  • Index
  • Greek symbols
  • Roman symbols
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