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Barycentric Interpolation

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Approximation Theory XIV: San Antonio 2013

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 83))

Abstract

This survey focusses on the method of barycentric interpolation, which ties up to the ideas that August Ferdinand Möbius published in his seminal work “Der barycentrische Calcul” in 1827. For univariate data, it leads to a special kind of rational interpolation which is guaranteed to have no poles and favorable approximation properties. We further discuss how to extend this idea to bivariate data, both for scattered data and for data given at the vertices of a polygon.

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Notes

  1. 1.

    Note that at least one of the \(w_i(x)\) must be negative if \(x\) is outside the convex hull of the nodes \(x_0,\ldots ,x_n\), which is physically impossible and motivates to call the \(w_i\) weights rather than masses.

  2. 2.

    Since \(n=m=1\), these are the unique barycentric basis functions, according to Möbius [24].

  3. 3.

    According to Henrici [17], this terminology goes back to Rutishauser [27] and is justified because the interpolating polynomial reproduces linear functions for \(n\ge 1\) and therefore is a barycentric interpolant.

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Authors and Affiliations

  1. Faculty of Informatics, Università della Svizzera italiana, Via Giuseppe Buffi 13, 6904, Lugano, Switzerland

    Kai Hormann

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  1. Kai Hormann
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Correspondence to Kai Hormann .

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Editors and Affiliations

  1. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois, USA

    Gregory E. Fasshauer

  2. Department of Mathematics, Vanderbilt University, Nashville, Tennessee, USA

    Larry L. Schumaker

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Hormann, K. (2014). Barycentric Interpolation. In: Fasshauer, G., Schumaker, L. (eds) Approximation Theory XIV: San Antonio 2013. Springer Proceedings in Mathematics & Statistics, vol 83. Springer, Cham. https://doi.org/10.1007/978-3-319-06404-8_11

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