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 $<f,g> = \CL(f g)$ 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.