On discrete orthogonal polynomials of several variables

Abstract

Let $V$ be a set of isolated points in \RR^d. Define a linear functional \CL on the space of real polynomials restricted on $V$, \CL f = \sum_{x \in V} f(x)\rho(x), where $\rho$ is a nonzero function on $V$. Polynomial subspaces that contain discrete orthogonal polynomials with respect to the bilinear form = \CL(f g) are identified. One result shows that the discrete orthogonal polynomials still satisfy a three-term relation and Favard's theorem holds in this general setting.Comment: 15 pages, 2 figure