Wavelet transforms versus Fourier transforms

Abstract

This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The "wavelet transform" maps each f(x) to its coefficients with respect to this basis. The mathematics is simple and the transform is fast (faster than the Fast Fourier Transform, which we briefly explain), but approximation by piecewise constants is poor. To improve this first wavelet, we are led to dilation equations and their unusual solutions. Higher-order wavelets are constructed, and it is surprisingly quick to compute with them -- always indirectly and recursively.