Bivariate spline functions and the approximation of linear functionals

The reproducing kernel for a Hilbert space of bivariate functions which have Taylor expansions is constructed. The concepts of optimal approximation of linear functionals in the sense of Sard and approximations resulting from bivariate spline functions are shown to be equivalent in these spaces. Bivariate splines that both smooth and interpolate are discussed.This research was supported by the Office of Naval Research under Grant NR 044-443.