On multivariate E-splines

Abstract

In a recent interesting paper, A. Ron [14] initiated the study of cardinal exponential splines in several variables. Beyond settling some open problems which were suggested by his results, this paper addresses a variety of additional questions of an algebraic nature. Thus some of the results we obtained elsewhere on cube spline surfaces appear now as a special case of the present general setting. Our approach is critically based on some new bounds and formulas for the dimension of certain intersections of null spaces of commuting linear operators where the particular way of forming these intersections is determined by a matroid structure. Specifically, this allows us to characterize the local linear independence of translates of the exponential cube spline as well as to study local algebraic properties of functions spanned by them. Only special cases were covered by Ron's investigation. Furthermore, we consider in detail the construction of dual bases, linear projectors, linear dependence relations among these exponential splines, and subdivision algorithms for their approximate evaluation. The analysis of these latter algorithms is based on the concept of discrete exponential splines. We investigate to what extent these discrete objects possess analogous properties to their continuous counterparts. Finally, the combinatorial interpretation of the discrete splines, their relationship to their continuous counterparts, and the machinery we develop leads to interesting combinatorial applications for enumerating the number of solutions to linear diophantine systems and corresponding reciprocity relations. An announcement of these results appeared in [10]. © 1989.