What is the natural generalization of univariate splines to higher dimensions?

• Authors: Marian Neamtu
• Venue: Mathematical Methods for Curves and Surfaces
• Year: 2001

In the first part of the paper, the problem of defining multivariate 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 difficulties 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 univariate splines. This generalization utilizes the novel concept of the so-called Delaunay configurations, 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.