The first constructions of smooth orthonormal wavelet bases seemed a bit miraculous, as illustrated by the proof in 4.2. A that the Meyer wavelets constitute an orthonormal basis. This situation changed with the advent of multiresolution analysis, formulated in the fall of 1986 by Mallat and Meyer. Multiresolution analysis provides a natural framework for the understanding of wavelet bases, and for the construction of new examples. The history of the formulation of multiresolution analysis is a beautiful example of applications stimulating theoretical development. When he first learned about the Meyer basis, Mallat was working on image analysis, where the idea of studying images simultaneously at different scales and comparing the results had been popular for many years (see, e.g., Witkin (1983) or Burt and Adelson (1983)). This stimulated him to view orthonormal wavelet bases as a tool to describe mathematically the "increment in information" needed to go from a coarse approximation to a higher resolution approximation. This insight crystallized into multiresolution analysis (Mallat (1989), Meyer (1986)).

5.1. The basic idea.

A multiresolution analysis consists of a sequence of successive approximation spaces V _j. More precisely, the closed subspaces V _j satisfy ^[1]

with

If we denote by P _j the orthogonal projection operator onto V _j, then (5.1.2) ensures that lim _{j ? ?} P _jf = f for all . There exist many ladders of spaces satisfying (5.1.1) (5.1.3) that have nothing to do with "multiresolution"; the multiresolution aspect is a consequence of the...