On discrete orthogonal polynomials of several variables

Let $V$ be a set of isolated points in ${\text{RR}}^d$. Define a linear functional $\text{CL}$ on the space of real polynomials restricted on $V$, $\CL f = \sum_{x \in V} f(x)\rho(x)$,where$\rho$isanonzerofunctionon$V$. Polynomial subspaces that contain discrete orthogonal polynomials with respect to the bilinear form $<f,g>=\text{CL}(fg)$ 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.