Multi-variable orthogonal polynomials

TL;DR

This paper characterizes atomic probability measures with finite atoms in multiple dimensions and explores the structure of Jacobi sequences for various multivariate orthogonal polynomials, revealing diagonal matrix properties.

Contribution

It provides a characterization of finite-atom measures and establishes the diagonal nature of Jacobi sequences for multivariate Hermite, Laguerre, and Jacobi polynomials, extending to Gegenbauer, Chebyshev, and Legendre cases.

Findings

Jacobi sequences for multiple Hermite, Laguerre, Jacobi polynomials are diagonal matrices.

Characterization of atomic probability measures with finitely many atoms.

Explicit Jacobi sequences for Gegenbauer, Chebyshev, and Legendre polynomials.

Abstract

We characterize the atomic probability measure on $\mathbb{R}^d$ which having a finite number of atoms. We further prove that the Jacobi sequences associated to the multiple Hermite (resp. Laguerre, resp. Jacobi) orthogonal polynomials are diagonal matrices. Finally, as a consequence of the multiple Jacobi orthogonal polynomials case, we give the Jacobi sequences of the Gegenbauer, Chebyshev and Legendre orthogonal polynomials.