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Approximation and Orthogonality in Sobolev Spaces on a Triangle

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Abstract

Approximation by polynomials on a triangle is studied in the Sobolev space \(W_2^r\) that consists of functions whose derivatives of up to r-th order have bounded \(L^2\) norm. The first part aims at understanding the orthogonal structure in the Sobolev space on the triangle, which requires explicit construction of an inner product that involves derivatives and its associated orthogonal polynomials, so that the projection operators of the corresponding Fourier orthogonal expansion commute with partial derivatives. The second part establishes the sharp estimate for the error of polynomial approximation in \(W_2^r\), when \(r = 1\) and \(r=2\), where the polynomials of approximation are the partial sums of the Fourier expansions in orthogonal polynomials of the Sobolev space.

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Acknowledgements

The author thanks Dr. Huiyuan Li for helpful discussions in early stage of this work, and thanks two anonymous referees for their helpful comments.

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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 Edward B. Saff.

The author was supported in part by NSF Grant DMS-1510296.

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Xu, Y. Approximation and Orthogonality in Sobolev Spaces on a Triangle. Constr Approx 46, 349–434 (2017). https://doi.org/10.1007/s00365-017-9377-3

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  • DOI: https://doi.org/10.1007/s00365-017-9377-3

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