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Bivariate spline interpolation with optimal approximation order

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Abstract

Let Δ be a triangulation of some polygonal domain Ω ⊂ R2 and let S rq (Δ) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to Δ. We develop the first Hermite-type interpolation scheme for S r q (Δ), q ≥ 3r + 2, whose approximation error is bounded above by Kh q +1, where h is the maximal diameter of the triangles in Δ, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and near-singular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of S r q (Δ). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines developed in [7] and [18].

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Authors and Affiliations

  1. Justus-Liebig-Universität Giessen, Mathematisches Institut, 35392, Giessen, Germany

    O. Davydov

  2. Universität Mannheim, Fakultät für Mathematik und Informatik, 68131, Mannheim, Germany

    G. Nürnberger & F. Zeilfelder

Authors
  1. O. Davydov
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  2. G. Nürnberger
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  3. F. Zeilfelder
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Communicated by T. N. T. Goodman.

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Davydov, O., Nürnberger, G. & Zeilfelder, F. Bivariate spline interpolation with optimal approximation order. Constr. Approx 17, 181–208 (2001). https://doi.org/10.1007/s003650010034

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  • DOI: https://doi.org/10.1007/s003650010034

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