Projection into Spaces Generated by Haar and Daubechies Scaling Functions

In this Demonstration, selects which wavelet scaling function is used to project a function into a collection of subspaces of . The approximation space is defined as the span of . Choices for are the Haar scaling function and the Daubechies scaling function with two vanishing moments.

Both scaling functions generate orthonormal bases. Thus the projection of into a space is given by , where .

For students: The function is nonzero on and the Haar scaling function is nonzero on . From this, determine what values of (as a function of ) make nonzero. What about for the Daubechies scaling function that is nonzero on ? You can check your answers by looking at the source code.

Scaling functions are the basic building blocks for multiresolution analysis in wavelet theory. For more information see [1]. This Demonstration is based on an example from that book.

Reference

[1] D. K. Ruch and P. J. Van Fleet, Wavelet Theory: An Elementary Approach with Applications, Hoboken, NJ: John Wiley & Sons, 2009.