Wavelet methods for pointwise regularity and local oscillations of functions
About this Title
Stéphane Jaffard and Yves Meyer
Publication: Memoirs of the American Mathematical Society
Publication Year 1996: Volume 123, Number 587
ISBNs: 978-0-8218-0475-9 (print); 978-1-4704-0172-6 (online)
MathSciNet review: 1342019
MSC (1991): Primary 42A16; Secondary 26A16, 26A30, 26B35
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.
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.
Table of Contents
- 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