Multivariate differences, polynomials, and splines

Abstract

We generalize the univariate divided difference to a multivariate setting by considering linear combinations of point evaluations that annihilate the null space of certain differential operators. The relationship between such a linear functional and polynomial interpolation resembles that between the divided difference and Lagrange interpolation. Applying the functional to the shifted multivariate truncated power produces a compactly supported spline by which the functional can be represented as an integral. Examples include, but are not limited to, the tensor product B-Spline and the box spline. © 1996 Academic Press, Inc.