Introduction

Abstract

Wavelet bases were introduced in the late 1980s as a tool for signal and image processing. Among the applications considered at the beginning we recall applications in the analysis of seismic signals, the numerous applications in image processing — image compression, edge-detection, denoising, applications in statistics, as well as in physics. Their effectiveness in many of the mentioned fields is nowadays well established: as an example, wavelets are actually used by the US Federal Bureau of Investigation (or FBI) in their fingerprint database, and they are one of the ingredients of the new MPEG media compression standard. Quite soon it became clear that such bases allowed to represent objects (signals, images, turbulent fields) with singularities of complex structure with a low number of degrees of freedom, a property that is particularly promising when thinking of an application to the numerical solution of partial differential equations: many PDEs have in fact solutions which present singularities, and the ability to represent such a solution with as little as possible degrees of freedom is essential in order to be able to implement effective solvers for such problems. The first attempts to use such bases in this framework go back to the late 1980s and early 1990s, when the first simple adaptive wavelet methods [32] appeared. In those years the problems to be faced were basic ones. The computation of integrals of products of derivatives of wavelets — objects which are naturally encountered in the variational approach to the numerical solution of PDEs — was an open problem (solved later by Dahmen and Michelli in [25]).