, Volume 36, Issue 2, pp 161-190
Date: 11 Jan 2012

Orthogonal Polynomials and Expansions for a Family of Weight Functions in Two Variables

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Orthogonal polynomials for a family of weight functions on [−1,1]2, $$\mathcal{W}_{\alpha,\beta,\gamma}(x,y) = |x+y|^{2\alpha+1}|x-y|^{2\beta+1} \bigl(1-x^2\bigr)^{\gamma}\bigl(1-y^2\bigr)^{\gamma},$$ 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 γ=±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.

Communicated by: Tom H. Koornwinder.