ABSTRACT
The next, and current, step in the development of Fourier analysis is to replace the classical sine and cosine building blocks with more flexible units—indeed, with units that can be tailored to the situation at hand. Such units should, ideally, be localizable; in this way they can more readily be tailored to any particular application. This is what wavelet theory is all about. This chapter shows how to develop a Multi-Resolution Analysis and how Fourier analysis may be carried out with localization in either the space variable or the Fourier transform variable; the reader can see how either variable may be localized. It describes the Haar wavelet basis, and presents two computational examples that provide concrete illustrations of how the Haar wavelet expansion is better behaved than the Fourier series expansion. The Haar wavelets are particularly effective at encoding information coming from a function that is constant on large intervals.
