Non - Existence of Star - supported

We consider polynomial spline spaces S r d (4) of degree d and smoothness r deened on triangulations. It is known that for d 3r + 2, S r d (4) possesses a basis of star-supported splines, i.e., splines whose supports are at most the set of triangles surrounding a vertex. Here we extend the theory by showing that for all d 3r + 1, there exist triangulations for which no such bases exist. x1. Introduction Given a regular triangulation 4, let S r d (4) := fs 2 C r (() : sj T 2 P d for all triangles T 2 4g; where P d is the space of polynomials of degree d, and is the union of the triangles in 4. Such spline spaces have been heavily studied, cf. e.g. 1{10] and references therein. Of particular interest for applications are spline spaces that possess a basis where every spline is supported only on the star of a vertex. (The star of a vertex is the set of triangles sharing that vertex.) Using such bases in applications leads to sparse linear systems. We call such splines star-supported. In 1] they are referred to as minimally supported, while in 5] they are called vertex splines. It is easy to see that for all d 1, the spaces S 0 d (4) have star-supported bases. In addition, for r 1, it is known 6,7] that the spaces S r d (4) possess bases of star-supported splines for all d 3r+2. The following complement to this result is the main result of this paper. Theorem 1.1. Suppose r 1 and d 3r + 1. Then there are triangulations 4 for which S r d (4) does not have a star-supported basis. Our proof of this theorem is based on showing that there exist triangulations such that the number of linearly independent star-supported splines in the space 1)