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

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.