Nonexistence of Star-Supported Spline Bases

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. 1. 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{14] and references therein.