The Haar System

In this project we will present an example of an orthonormal system on [0,1) known asrnthe Haar system. The Haar basis is the simplest and historically the first example of anrnorthonormal wavelet basis. Many of its properties stand in sharp contrast to therncorresponding properties of the trigonometric basis. For example,rn(1) The Haar basis functions are supported on small subintervals of [0, 1),rnwhereas the Fourier basis functions are nonzero on all of [0,1);rn(2) The Haar basis functions are step functions with jump discontinuities,rnwhereas the Fourier basis functions are __ on [0, 1);rn(3) The Haar basis replaces the notion of frequency (represented by the index nrnin the Fourier basis) with the dual notions of scale and location (separatelyrnindexed by j and k); andrn(4) The Haar basis provides a very efficient representation of functions thatrnconsist of smooth, slowly varying segments punctuated by sharp peaks andrndiscontinuities, whereas the Fourier basis best represents functions thatrnexhibit long term oscillatory behavior.

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