Wavelets, Vibrations and Scalings

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Providence 1999 Hardcover First Edition. 131 pages. Hardcover, no dustjacket. Brand new copy. MATHEMATICS. A mathematical treatise on fractal sets and extremely irregular ... functions, and the growing role they are playing. Includes Index. (Key Words; Fractals, Mathematics, Wavelets, Vibrations, Scalings, Microlocal Analysis, Zygmund Class ). Read more Show Less

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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 A pioneer of the wavelet ''saga'', Meyer gives new and as yet unpublished results throughout the book.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.

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Editorial Reviews

Discusses the relevance of wavelet analysis to problems in which self- similarities are important, concluding that a weak form of self- similarity can be given a simple characterization through size estimates on wavelet coefficients, and that wavelet bases can be tuned in order to provide a sharper characterization of this self- similarity. Contains chapters on scaling exponents at small scales, infrared divergences and Hadamard's finite parts, 2-microlocal spaces and new characterizations, an adapted wavelet basis, and combining a Wilson basis with a wavelet basis. For graduate students, research mathematicians, physicists, and others working in wavelet analysis and applications in multifractal signal processing. Annotation c. by Book News, Inc., Portland, Or.
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Product Details

  • ISBN-13: 9780821806852
  • Publisher: American Mathematical Society
  • Publication date: 11/18/1997
  • Series: Crm Monograph Series, #9
  • Pages: 133

Table of Contents

List of Figures
Introduction 1
Ch. 1 Scaling exponents at small scales 5
Ch. 2 Infrared divergences and Hadamard's finite parts 43
Ch. 3 The 2-microlocal spaces [actual symbol not reproducible] 57
Ch. 4 New characteristics of the two-microlocal spaces 79
Ch. 5 An adapted wavelet basis 89
Ch. 6 Combining a Wilson basis with a wavelet basis 111
Bibliography 127
Index 129
Greek Symbols 131
Roman Symbols 133
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  • Anonymous

    Posted January 9, 2003

    multifractal signals

    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 lets the reader look behind the curtain of the abstract math, or the technicalities of the algorithms; giving lucid explanation of the key concepts,-- where they come from, and how they are used. Central concepts in the book: Holder exponent, scaling exponent, Hausdorff dimension, microlocal spaces, homogeneous distributions, Littlewood-Paley expansion, cone of influence, Wilson basis, Heisenberg [time-frequency] boxes,...

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