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.

Readership

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

Table of Contents

• Introduction
• Modulus of continuity and two-microlocalization
• Singularities of functions in Sobolev spaces
• Wavelets and lacunary trigonometric series
• Properties of chirp expansions
• Trigonometric chirps
• Logarithmic chirps
• The Riemann series
• References
• Index
• Notations