Since their introduction in the 1980's, wavelets have become a powerful tool in mathematical analysis, with applications such as image compression, statistical estimation and numerical simulation of partial differential equations. One of their main attractive features is the ability to accurately represent fairly general functions with a small number of adaptively chosen wavelet coefficients, as well as to characterize the smoothness of such functions from the numerical behaviour of these coefficients. The theoretical pillar that underlies such properties involves approximation theory and function spaces, and plays a pivotal role in the analysis of wavelet-based numerical methods.
Themes from math, science and engineering that predate the more recent trends in wavelts are signal processing, atomic decomposions of Calderon, splines, and finite element techniques of numerical analysis. All of these earlier ideas have enriched math, given new life and power to applied wavelet technology. While signal processing, filter banks, and pyramid algorithms have formed the under-current of a number of recent books on the interface of wavelets and some the neighboring fields, including applications, the interaction with numerical analysis hasn't really been presented in a systematic fashion in book form; so that students, and the rest of us, can pick and learn the fundamentals from scratch;-- not before this lovely new book by Albert Cohen: You find the methods and tools needed in the discretization of problems from PDE and from analysis. I hope the publisher will promote the book. It deserves it. Give it a try! I believe you will not be disappointed. Albert Cohen shows how wavelet tools have enriched numerical analysis and vice versa.
