One-Dimensional Wavelet Systems

Abstract

In this chapter, we introduce the basic scaling and wavelet functions of wavelet analysis, using the algebraic tools developed in the previous chapter. The principal result is that for any wavelet matrix A ∈ WM (m, g ; C), there is a scaling function ϕ(x) and m — 1 wavelet functions ψ¹(x), …, ψ^m−1(x) which satisfy specific scaling relations defined in terms of the wavelet matrix A. These functions are all compactly supported and square-integrable, and their rescalings and translations, called the wavelet system determined by A, provide a basis for L² (R). Generically, the wavelet system will be an orthonormal basis, but in all cases, even when orthogonality is not present, these functions will allow reconstruction of a given f ∈ L² (R) from the scalar products of f with the elements of the wavelet system, just as if it had been an orthonormal basis (this is the tight frame property of wavelets, discussed in Section 5.3.1).