Cascade algorithm

Summary

The cascade algorithm computes the scaling function of a wavelet from its filter coefficients by iterating the refinement equation. It starts from a simple function, such as the indicator of $[0,1)$. Each step builds a new function from shrunken and shifted copies of the old one. For good filters, such as the filters of the Daubechies wavelets, the iterates converge to the scaling function. Most compactly supported wavelets have no closed formula, so this is the usual way to draw them.