Wavelets

Abstract

In this chapter we develop an important variation on Fourier series, replacing the sine and cosine functions with new families of functions, called wavelets. The strategy is to construct wavelets so that they have some of the good properties of trig functions but avoid the failings of Fourier series that we have seen in previous chapters. With such functions, we can develop new versions of Fourier series methods that will work well for problems where traditional Fourier series work poorly.

What are the good properties of trig functions? First and foremost, we have an orthogonal basis in L ², namely the set of functions sin(nx) and cos(nx) as n runs over ${\mathbb{N}_0}$. This leads to the idea of breaking up a wave into its harmonic constituents, as the sine and cosine functions appear in the solution of the wave equation. We want to retain some version of this orthogonality.