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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