Constructive Approximation

, Volume 36, Issue 2, pp 161–190 | Cite as

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

Article

Abstract

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.

Keywords

Orthogonal polynomials Orthogonal expansions Jacobi polynomials Two variables Lebesgue constants 

Mathematics Subject Classification (2000)

33C50 42C10 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OregonEugeneUSA

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