Quadratic decomposition of bivariate orthogonal polynomials

TL;DR

This paper introduces a quadratic decomposition framework for bivariate orthogonal polynomials, linking polynomial sequences through Christoffel transformations and matrix transformations, with applications to polynomials on the ball and simplex.

Contribution

It presents a novel quadratic decomposition method for bivariate orthogonal polynomials, incorporating Christoffel transformations and Backlund type matrix transformations.

Findings

Decomposition relates orthogonal polynomials via quadratic transformations.

Connections established between polynomials on the ball and the simplex.

Framework facilitates construction of symmetric bivariate orthogonal polynomials.

Abstract

We describe bivariate polynomial sequences orthogonal to a symmetric weight function in terms of several bivariate polynomial sequences orthogonal with respect to Christoffel transformations of the initial weight under a quadratic transformation. We analyze the construction of a symmetric bivariate orthogonal polynomial sequence from a given one, orthogonal to a weight function defined on the positive plane. In this description plays an important role a sort of Backlund type matrix transformations for the involved three term matrix coefficients. We take as a case study relations between symmetric orthogonal polynomials defined on the ball and on the simplex.