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Error Bounds for Minimal Energy Bivariate Polynomial Splines

By Manfred Von Golitschek, Ming-Jun Lai and Larry L. Schumaker


. We derive error bounds for bivariate spline interpolants which are calculated by minimizing certain natural energy norms. x1. Introduction Suppose we are given values ff(v)g v2V of an unknown function f at a set V of scattered points in IR 2 . To approximate f , we choose a linear space S of polynomial splines of degree d defined on a triangulation 4 with vertices at the points of V. Let U f := fs 2 S : s(v) = f(v); v 2 Vg (1:1) be the set of all splines in S that interpolate f at the points of V. We assume that S is big enough so that U f is nonempty. Then a commonly used way to create an approximation of f (cf. [6--10]) is to choose a spline S f such that E(S f ) = min s2Uf E(s); (1:2) where E(s) := X T24 Z T \Theta s 2 xx + 2s 2 xy + s 2 yy : (1:3) We refer to S f as the minimal energy interpolating spline. 1) Institut fur Angewandte Mathematik und Statistik der Universitat Wurzburg, 97074 Wurzburg, Germany, 2) Departmen..

Year: 1999
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