# Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions

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*St(’)ephane Jaffard; Yves Meyer*

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.

#### Table of Contents

# Table of Contents

## Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions

- Table of contents vii8 free
- Introduction 112 free
- I. Modulus of continuity and two-microlocalization 819 free
- II. Singularities of functions in Sobolev spaces 3041
- III. Wavelets and lacunary trigonometric series 4859
- IV. Properties of chirp expansions 5667
- V. Trigonometric chirps 7283
- VI. Logarithmic chirps 89100
- VII. The Riemann series 96107
- References 106117
- Index 109120 free
- Notations 110121