On some properties of the entries of a refinement equation and of their generalizations

The entries of the matrix coefficients of the refinement equation corresponding to the multiresolution analysis of Alpert are actually connection coefficients, which turns out to be related to a wide range of problems. We’ll start by discussing some history and then a generalization of a discrete wave equation that is satisfied by connection coefficients will be presented. After that, we’ll consider some simulations of the behavior of the entries, which are in fact functions of two discrete variables and they can be expressed in terms of the Legendre polynomials. Next, we’ll generalize these functions to the case of the ultraspherical polynomials and show that these new functions obey two generalized eigenvalue problems in each of the two discrete variables, which constitute a generalized bispectral problem. At the end, we’ll make some connections to spectral transformations. The talk is based on a joint work with Jeff Geronimo