Approximating Continuous Functions with Haar Approximations

The Haar scaling function is defined by

On [0,1) 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.

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."