Orthogonal polynomials in several variables for measures with mass points

Abstract

Let $dν$ be a measure in $\mathbb{R}^d$ obtained from adding a set of mass points to another measure $dμ$. Orthogonal polynomials in several variables associated with $dν$ can be explicitly expressed in terms of orthogonal polynomials associated with $dμ$, so are the reproducing kernels associated with these polynomials. The explicit formulas that are obtained are further specialized in the case of Jacobi measure on the simplex, with mass points added on the vertices, which are then used to study the asymptotics kernel functions for $dν$.