Combinatorics for Wavelets: The Umbral Refinement Equation

The refinement equation is the most fundamental equation in wavelet theory.In this article, we study its combinatorial meanings and analogs. We show that the invariant properties of the Cauchy generating function are well adapted to the translation and dilation operators in the refinement equation. This leads to the discovery of analytic scaling functions and wavelets. The classical umbral calculus (or symbolic calculus) provides a powerful tool for moments analysis and defines the combinatorial analog of the ordinary refinement equation—the umbral refinement equation. By developing the existing theory of classical umbral calculus, we are able to solve the umbral refinement equation in a purely umbral manner. Many classical results in wavelet analysis are reestablished in the context of umbral calculus.