Orthogonal polynomials and Gaussian quadrature for refinable weight functions

Abstract

We show how to compute the modified moments of a refinable weight function directly from its mask in O ( N 2 n ) rational operations, where N is the desired number of moments and n the length of the mask. Three immediate applications of such moments are: • the expansion of a refinable weight function as a Legendre series; • the generation of the polynomials orthogonal with respect to a refinable weight function; • the calculation of Gaussian quadrature formulas for refinable weight functions. In the first two cases, all operations are rational and can in principle be performed exactly.