Orthogonal polynomials and expansions for a family of weight functions in two variables

TL;DR

This paper investigates orthogonal polynomials associated with a specific family of weight functions in two variables, establishing their relation to Koornwinder polynomials, providing explicit bases for special cases, and analyzing convergence properties.

Contribution

It introduces a new family of orthogonal polynomials related to Koornwinder polynomials, with explicit bases for certain parameters and a study of their expansion convergence.

Findings

Explicit orthogonal polynomial bases for special parameter values

Closed-form reproducing kernel formulas

Convergence analysis of orthogonal expansions

Abstract

Orthogonal polynomials for a family of weight functions on $[-1,1]^2$, $$ \CW_{\a,\b,\g}(x,y) = |x+y|^{2\a+1} |x-y|^{2\b+1} (1-x^2)^\g(1-y^2)^{\g}, $$ are studied and shown to be related to the Koornwinder polynomials defined on the region bounded by two lines and a parabola. In the case of $\g = \pm 1/2$, an explicit basis of orthogonal polynomials is given in terms of Jacobi polynomials and a closed formula for the reproducing kernel is obtained. The latter is used to study the convergence of orthogonal expansions for these weight functions.