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Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions

St(’)ephane Jaffard CERMA-ENPC
Yves Meyer University of Paris IX
Available Formats:
Electronic ISBN: 978-1-4704-0172-6
Product Code: MEMO/123/587.E
110 pp
List Price: $45.00 MAA Member Price:$40.50
AMS Member Price: $27.00 Click above image for expanded view Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions St(’)ephane Jaffard CERMA-ENPC Yves Meyer University of Paris IX Available Formats:  Electronic ISBN: 978-1-4704-0172-6 Product Code: MEMO/123/587.E 110 pp  List Price:$45.00 MAA Member Price: $40.50 AMS Member Price:$27.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 1231996
MSC: Primary 26; 42;

Currently, new trends in mathematics are emerging from the fruitful interaction between signal processing, image processing, and classical analysis.

One example is given by “wavelets”, which incorporate both the know-how of the Calderon-Zygmund school and the efficiency of some fast algorithms developed in signal processing (quadrature mirror filters and pyramidal algorithms.)

A second example is “multi-fractal analysis”. The initial motivation was the study of fully developed turbulence and the introduction by Frisch and Parisi of the multi-fractal spectrum. Multi-fractal analysis provides a deeper insight into many classical functions in mathematics.

A third example—“chirps”—is studied in this book. Chirps are used in modern radar or sonar technology. Once given a precise mathematical definition, chirps constitute a powerful tool in classical analysis.

In this book, wavelet analysis is related to the 2-microlocal spaces discovered by J. M. Bony. The authors then prove that a wavelet based multi-fractal analysis leads to a remarkable improvement of Sobolev embedding theorem. In addition, they show that chirps were hidden in a celebrated Riemann series.

Features:

• Provides the reader with some basic training in new lines of research.
• Clarifies the relationship between pointwise behavior and size properties of wavelet coefficents.

Graduate students and researchers in mathematics, physics, and engineering who are interested in wavelets.

• Chapters
• Introduction
• I. Modulus of continuity and two-microlocalization
• II. Singularities of functions in Sobolev spaces
• III. Wavelets and lacunary trigonometric series
• IV. Properties of chirp expansions
• V. Trigonometric chirps
• VI. Logarithmic chirps
• VII. The Riemann series
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Volume: 1231996
MSC: Primary 26; 42;

Currently, new trends in mathematics are emerging from the fruitful interaction between signal processing, image processing, and classical analysis.

One example is given by “wavelets”, which incorporate both the know-how of the Calderon-Zygmund school and the efficiency of some fast algorithms developed in signal processing (quadrature mirror filters and pyramidal algorithms.)

A second example is “multi-fractal analysis”. The initial motivation was the study of fully developed turbulence and the introduction by Frisch and Parisi of the multi-fractal spectrum. Multi-fractal analysis provides a deeper insight into many classical functions in mathematics.

A third example—“chirps”—is studied in this book. Chirps are used in modern radar or sonar technology. Once given a precise mathematical definition, chirps constitute a powerful tool in classical analysis.

In this book, wavelet analysis is related to the 2-microlocal spaces discovered by J. M. Bony. The authors then prove that a wavelet based multi-fractal analysis leads to a remarkable improvement of Sobolev embedding theorem. In addition, they show that chirps were hidden in a celebrated Riemann series.

Features:

• Provides the reader with some basic training in new lines of research.
• Clarifies the relationship between pointwise behavior and size properties of wavelet coefficents.