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Approximating Continuous Functions with Haar Approximations

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The Haar scaling function is defined by

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On the Haar space is spanned by , where is a non-negative integer. A continuous function can be approximated by a piecewise constant function by projection into Haar space.

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Contributed by: Sijia Liang and Bruce Atwood  (July 2011)
(Beloit College)
After work by: Helmut Knaust
Open content licensed under CC BY-NC-SA

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Since the basis functions form an orthonormal set, the approximation of in is , where is determined by the inner product . Students should ask themselves, "What is the interpretation of the projection of into ?" For more information, see Hemut Knaust, "Multi-Resolution Analysis for the Haar Wavelet."

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