Grid Point Interpolation on Finite Regions Using C1 Box Splines

Multivariate grid point interpolation on finite regions is considered by translates of C1 box splines defined on a (s + 1)-direction mesh in Rs. In general, this problem will give more degrees of freedom than the number of interpolation conditions, and hence the problem has no unique solution. Among all interpolants, the one minimizing a smoothing functional is chosen. Two choices of the smoothing functional are proposed, and the related existence and uniqueness problems are studied. Examples showing constructions of box spline surfaces on nonrectangular regions in R2 and rectangular regions in R3 are presented. The method can also be used to construct a smooth interpolant to a set of points given at an almost arbitrary set of grid points in Rs.