Overview

The triumphant progress wavelets have made in a great variety of applications is based in the first place on the so-called "fast algorithms" (fast wavelet transform, FWT), and these in turn owe their existence to a careful choice of the mother wavelet ?. So far in this book the particular mother wavelet chosen only had to fulfill some "technical" conditions, such as t ^r ? ? L ¹ or ? ? C ^r for some r ? 0 and, of course, or, even better, ? should be of a certain order N > 1.

The trigonometric basis functions e _?: t ? e ^{i ? t} are distinguished by the following linear reproducing property: If such a function is subject to a transition T _h, it simply picks up a constant factor:

Contrary to this, in the realm of wavelets the operation of scaling is the central theme, i.e., for arbitrary a ? ? ^* the operation

With respect to this operation, the wavelets considered so far did not behave in a special way (except ? _Haar). OK, their graph became flattened out or got compressed in the t-direction, depending on the value of a, but there was no reproduction property in the sense that the scaled version of a ? could be related to the original ? in some other way. In the discrete case only the integer iterates of...