2.1 Introduction

Time-frequency wavelets, which began with work by Dennis Gabor and by John von Neumann in the late 1940s, have a relatively long history in signal processing. Many of the fundamental contributions were subsequently achieved by physicists, and here we are thinking of work by Francis Low, Roger Balian, and Kenneth Wilson. Time-frequency wavelets have been widely used in speech processing, as will be shown in Chapter 6. Mathematicians did not pay much attention to this field. In contrast, the use of time-scale wavelets for signal and image processing is relatively recent, dating from the 1980s. However, in looking back over the history of mathematics, we will uncover at least seven different origins of wavelet analysis. Most of this work was done around the 1930s, and at that time the separate efforts did not appear to be parts of a coherent theory. Only today do we see how this work fits into the history of the theory of wavelets.

We feel that it is important to describe these sources in some detail. Each of them corresponds to a specific point of view and a particular technique, which only now are we able to view from a common scientific perspective. What's more, these specific techniques were rediscovered several years ago by physicists and mathe-maticians working on wavelets. Matthias Holschneider used, without knowing it, Lusin's technique (1930) to analyze Riemann's function (sections 2.6 and 10.4). Grossmann and Morlet rediscovered Alberto Calder n's identity (1960) 20 years later. And to spare no one,...