Orthogonal polynomials in several variables for measures with mass points

TL;DR

This paper derives explicit formulas for multivariable orthogonal polynomials with measures that include mass points, and studies their asymptotic kernel functions, especially for Jacobi measures on simplices with vertex mass points.

Contribution

It provides explicit expressions for orthogonal polynomials and kernels in multiple variables with added mass points, extending classical results to more complex measures.

Findings

Explicit formulas for orthogonal polynomials with mass points

Specialized formulas for Jacobi measures on simplices

Asymptotic analysis of kernel functions with added mass points

Abstract

Let $d\nu$ be a measure in $\mathbb{R}^d$ obtained from adding a set of mass points to another measure $d\mu$. Orthogonal polynomials in several variables associated with $d\nu$ can be explicitly expressed in terms of orthogonal polynomials associated with $d\mu$, 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 $d\nu$.