Multivariate refinable functions of high approximation order via quotient ideals of Laurent polynomials

Abstract

We give an algebraic interpretation of the well-known “zero-condition” or “sum rule” for multivariate refinable functions with respect to an arbitrary scaling matrix. The main result is a characterization of these properties in terms of containment in a quotient ideal, however not in the ring of polynomials but in the ring of Laurent polynomials.