Bivariate spline interpolation with optimal approximation order O. Davydov, G. Nürnberger and F. Zeilfelder, Bivariate spline interpolation with optimal approximation order, Constr. Approx. 17 (2001), 181-208.

Abstract: Let $\Delta$ be a triangulation of some polygonal domain $\Omega\subset\RR^2$ and let $S_q^r(\Delta)$ denote the space of all bivariate polynomial splines of smoothness $r$ and degree $q$ with respect to $\Delta$. We develop the first Hermite type interpolation scheme for $S_q^r(\Delta)$, $q\ge 3r+2$, whose approximation error is bounded above by $Kh^{q+1}$, where $h$ is the maximal diameter of the triangles in $\Delta$, 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 $S_q^r(\Delta)$. 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 [7,18].

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