BIVARIATE SPLINE INTERPOLATION AT GRID POINTS

We describe algorithms for constructing point sets at which interpolat ion by spaces of bivariate splines of arbitrary degree and smoothness is possible. The splines are defined on rectangular partitions adding one or two diagonals to each rectangle. The interpolation sets are sel ected in such a way that the grid points of the partition are containe d in these sets, and no large linear systems have to be solved. Our me thod is to generate a net of line segments and to choose point sets in these segments which satisfy the Schoenberg-Whitney condition for cer tain univariate spline spaces such that a principle of degree reductio n can be applied. In order to include the grid points in the interpola tion sets, we give a sufficient Schoenberg-Whitney type condition for interpolation by bivariate splines supported in certain cones. This ap proach is completely different from the known interpolation methods fo r bivariate splines of degree at most three. Our method is illustrated by some numerical examples.