Error Bounds for Minimal

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 deened on a triangulation 4 with vertices at the points of V. 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. 8{12]) is to choose a spline S f such that E(S f) = min s2U f E(s); (1:2) where E(s) := X T24 Z T s 2 xx + 2s 2 xy + s 2 yy : (1:3) We refer to S f as the minimal energy interpolating spline.