The Haar System
In this chapter we will present an example of an orthonormal system on [0,1] known as the Haar system. The Haar basis is the simplest and historically the first example of an orthonormal wavelet basis. Many of its properties stand in sharp contrast to the corresponding properties of the trigonometric basis (Definition 2.5). For example, (1) the Haar basis functions are supported on small subintervals of [0,1], whereas the Fourier basis functions are nonzero on all of [0,1]; (2) the Haar basis functions are step functions with jump discontinuities, whereas the Fourier basis functions are C ∞ on [0,1]; (3) the Haar basis replaces the notion of frequency (represented by the index n in the Fourier basis) with the dual notions of scale and location (separately indexed by j and k); and (4) the Haar basis provides a very efficient representation of functions that consist of smooth, slowly varying segments punctuated by sharp peaks and discontinuities, whereas the Fourier basis best represents functions that exhibit long term oscillatory behavior. More will be said about this contrast in Section 5.4.
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