Multivariate B-splines on triangulated rectangles

Abstract

Although many results in univariate spline theory have been extended to the higher-dimensional settings by taking tensor products, very little is known on the general theory of multivariate spline functions. Since a univariate spline function is a “smooth” piecewise polynomial separated by a set of points which are called knots, a bivariate spline function is a “smooth” piecewise polynomial in two variables separated by a grid of curves, and so on. In the two-dimensional setting, for example, if a domain G’ in IFi’ is divided into a finite or countable number of cells by a grid partition A, then the space S;(A) of multivariate (or, more precisely, bivariate) spline functions is the collection of all functions in P(U) such that the restriction of every s E S:(A) to each cell of the grid partition is a polynomial p(.u, 4’) of total degree k, namely,