Examples of One-Dimensional Wavelet Systems

Abstract

The theory of compactly supported wavelets has produced many new classes of orthonormal basis functions. This chapter collects a number of examples of scaling functions and wavelets that have helped the authors to understand wavelets and develop intuition about them. The classes of wavelets in the list below seem to us to capture many of the features that are important for applications or have a special signficance for the mathematical theory. Some of the examples listed here are treated in more detail in other chapters. The examples are designed to show properties that are shared by wavelets of different rank or genus, and also to identify some properties that explicitly depend on these characteristic constants. With the exception of the sinc scaling and wavelet functions (which are universal), we include in our examples only compactly supported wavelets, which is the major thrust of this book. There are many examples of noncompactly supported wavelets with varying degrees of growth at infinity (polynomial, exponential decay, etc.), and we refer to the books by Chui [23], Daubechies [40], and Meyer [125], which discuss some of them in much more detail.