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

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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.

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Acknowledgements

The author thanks an anonymous referee for his careful review. The work was supported in part by NSF Grant DMS-1106113.

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Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403-1222, USA

    Yuan Xu

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  1. Yuan Xu
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Correspondence to Yuan Xu.

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Communicated by: Tom H. Koornwinder.

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Xu, Y. Orthogonal Polynomials and Expansions for a Family of Weight Functions in Two Variables. Constr Approx 36, 161–190 (2012). https://doi.org/10.1007/s00365-011-9149-4

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  • DOI: https://doi.org/10.1007/s00365-011-9149-4

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