Bivariate spline interpolation with optimal approximation order

Let be a triangulation of some polygonal domain f c R2 and let S9 (A) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to A. We develop the first Hermite-type interpolation scheme for S9 (A), q >_ 3r + 2, whose approximation error is bounded above by Kh4+i, where h is the maximal diameter of the triangles in A, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and nearsingular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of Sr, (A). 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 [71 and [181

Menopausal hormone therapy and 20-year breast cancer mortality

We give a survey of recent methods to construct Lagrange interpolation points for splines of arbitrary smoothness rand degree q on general crosscut partitions in IR2. For certain regular types of partitions, also results on Hermite interpolation sets and on the approximation order of the corresponding interpolating splines are given

Interpolation by splines on triangulations

We review recently developed methods of constructing Lagrange and Hermite interpolation sets for bivariate spline on triangulations of general type. Approximation order and numerical performance of our methods are also discussed