9 - Wavelet methods with radial basis functions

Summary

Introduction to wavelets and prewavelets

Already in the previous chapter we have discussed in what cases L²-approximants or other smoothing methods such as quasi-interpolation or smoothing splines with radial basis functions are needed and suitable for approximation in practice, in particular when data or functions f underlying the data are at the beginning not very smooth or must be smoothed further during the computation. The so-called wavelet analysis that we will introduce now is a further development in the general context of L²-methods, and indeed everything we say here will concern L²-functions, convergence in the L²-norm etc. only. Many important books have been written on wavelets before, and since this is not at all a book on wavelets, we will be fairly short here. The reader who is interested in the specific theory of wavelets is directed to one of the excellent works on wavelets mentioned in the bibliography, for instance the books by Chui, Daubechies, Meyer and others. Here, our modest goal is to describe what wavelets may be considered as in the context of radial basis functions. The radial basis functions turn out to be useful additions to the theory of wavelets because of the versatility of the available radial basis functions.

Given a square-integrable function f on ℝ, say, the aim of wavelet analysis is to decompose it simultaneously into its time and its frequency components.