**CRM Monograph Series**

Volume: 9;
1998;
133 pp;
Hardcover

MSC: Primary 26; 42;

**Print ISBN: 978-0-8218-0685-2
Product Code: CRMM/9**

List Price: $41.00

AMS Member Price: $32.80

MAA Member Price: $36.90

**Electronic ISBN: 978-1-4704-3855-5
Product Code: CRMM/9.E**

List Price: $38.00

AMS Member Price: $30.40

MAA Member Price: $34.20

# Wavelets, Vibrations and Scalings

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*Yves Meyer*

A co-publication of the AMS and Centre de Recherches Mathématiques

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unpublished results throughout the book.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 & Endorsements

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

# Table of Contents

## Wavelets, Vibrations and Scalings

- Cover Cover11
- Title page i2
- Contents iii4
- List of figures v6
- Préface vii8
- Introduction 110
- Scaling exponents at small scales 514
- Infrared divergences and Hadamard’s finite parts 4352
- The 2-microlocal spaces 𝐶^{𝑠,𝑠′}_{𝑥₀} 5766
- New characterizations of the two-microlocal spaces 7988
- An adapted wavelet basis 8998
- Combining a Wilson basis with a wavelet basis 111120
- Bibliography 127136
- Index 129138
- Greek symbols 131140
- Roman symbols 133142
- Back Cover Back Cover1145