The Haar System
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Date
2012-01-01
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Addis Ababa University
Abstract
In this project 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. 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 __ 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.
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Keywords
Haar System, Dyadic Step Functions, Dyadic Intervals, Comparison of Haar