What Is the Natural Generalization of Univariate Splines to Higher Dimensions?

In the rst part of the paper, the problem of deening multi-variate splines in a natural way is formulated and discussed. Then, several existing constructions of multivariate splines are surveyed, namely those based on simplex splines. Various diiculties and practical limitations associated with such constructions are pointed out. The second part of the paper is concerned with the description of a new generalization of univari-ate splines. This generalization utilizes the novel concept of the so-called Delaunay conngurations, used to select collections of knot-sets for simplex splines. The linear span of the simplex splines forms a spline space with several interesting properties. The space depends uniquely and in a local way on the prescribed knots and does not require the use of auxiliary or perturbed knots, as is the case with some earlier constructions. Moreover, the spline space has a useful structure that makes it possible to represent polynomials explicitly in terms of simplex splines. This representation closely resembles a familiar univariate result in which polar forms are used to express polynomials as linear combinations of the classical B-splines. x1. Introduction This paper describes the material I presented at the Fifth International Conference on Mathematical Methods for Curves and Surfaces, held in Oslo in June 2000. In my talk, I addressed the topic of a meaningful generalization of the classical univariate splines to higher dimensions. The quest for nding such generalizations is certainly not new. Many researchers in the theory of multivariate splines have addressed this or similar questions. Indeed, there is a great variety of multivariate generalizations of splines available today. However , there does not seem to exist a generalization that is commonly agreed to be the \right" one. This fact perhaps prompted Carl de Boor to conclude his survey \What is a multivariate spline?" 8] with a touch of irony: \If this leaves you a bit wondering what multivariate splines might be, I am pleased. For I don't know myself." Given the large number of possible approaches to multivariate splines, it is clear that the term \natural generalization" in the title is necessarily vague Mathematical Methods for Curves and Surfaces: Oslo 2000 355 Tom Lyche and Larry L. Schumaker (eds.), pp. 355{392. All rights of reproduction in any form reserved.