Orthogonal polynomials and diffusion operators

TL;DR

This paper classifies domains and measures in R^d where symmetric second order differential operators can be expanded on orthogonal polynomial bases, identifying 11 compact 2D cases and some non-compact ones.

Contribution

It provides a classification of domains and measures allowing orthogonal polynomial expansions for certain diffusion operators in two dimensions.

Findings

Identifies 11 compact domains in 2D for such operators

Provides examples of non-compact cases in 2D

Establishes conditions for orthogonal polynomial expansions

Abstract

We want to describe the triplets (\Omega, (g), \mu) where (g) is the (co)metric associated to some symmetric second order differential operator L defined on the domain \Omega of R^d and such that L is expandable on a basis of orthogonal polynomials of L_2(\mu), and \mu is some admissible measure. Up to affine transformation, we find 11 compact domains in dimension 2, and also give some non--compact cases in this dimension.