Darboux transformations for multivariate orthogonal polynomials

TL;DR

This paper extends Darboux transformations to multivariate orthogonal polynomials, providing a multidimensional Christoffel formula involving quasi-determinants and sample matrices, with conditions for poised sets discussed.

Contribution

It introduces a multidimensional Christoffel formula for orthogonal polynomials using Darboux transformations and analyzes conditions for poised sets in complex affine space.

Findings

Derived multivariate Darboux transformations for polynomial perturbations

Extended Christoffel formula to multiple dimensions using quasi-determinants

Discussed algebraic conditions for the existence of poised sets

Abstract

Darboux transformations for polynomial perturbations of a real multivariate measure are found. The 1D Christoffel formula is extended to the multidimensional realm: multivariate orthogonal polynomials are expressed in terms of last quasi-determinants and sample matrices. The coefficients of these matrices are the original orthogonal polynomials evaluated at a set of nodes, which is supposed to be poised. A discussion for the existence of poised sets is given in terms of algebraic hypersufaces in the complex affine space.