Wavelets: Theory and Applications

Introduction

This book introduces an integral transform and its discrete forms, which in recent years have delved into new areas of analysis and synthesis of signals: pattern recognition, as well as data compression [Ma187, Ma189b, ZHL90], numerical analysis [BCR91, DK92, Jaf92, XS92, DPS94, DPS93, Kun94], quantum field theory [BF87] and acoustics [GHKMM87, KMMG87] are just a few examples of many such areas.

Wavelet theory originates from signal theory. In 1984, Goupillaud, Grossman and Morlet published a paper [GGM85] in which they introduced a new transform for the frequency analysis of signals (time-dependent functions) and discussed results this enabled them to deduce. This new transform, known in the interim as the wavelet transform, was introduced because the classical methods of frequency analysis, i.e. Fourier and windowed Fourier transforms, have considerable disadvantages as regards signal theory. In mathematical circles, the continuous wavelet transform had already been known for some time as Calderón’s reproducing formula [Ca164], cf. also David [Dav91] and Meyer [Mey93]. However, its breakthrough set in with the appearance of the paper [GGM85] and the development of a discrete variant.

One of the major disadvantages of the Fourier transform lies in its lack of localization: the Fourier transform considers phenomena in an infinite interval, and this is very far from our everyday point of view. It decomposes a signal in plane waves (trigonometric functions), which oscillate ...