 
Summary: c
fl000 Society for Industrial and Applied Mathematics
Vol. 000, No. 000, 000 000
NONEXISTENCE OF STARSUPPORTED SPLINE BASES *
PETER ALFELD y AND LARRY L. SCHUMAKER z
Abstract. We consider polynomial spline spaces S r
d
(4) of degree d and smoothness r defined
on triangulations. It is known that for d – 3r + 2, S r
d
(4) possesses a basis of starsupported 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 Ÿ 3r + 1, there exist triangulations for which no such bases exist.
Key words. multivariate splines, piecewise polynomial functions, triangulations
AMS subject classifications. 41A63, 41A15, 65D07
1. Introduction. Given a regular triangulation 4, let
S r
d (4) := fs 2 C
r(\Omega\Gamma : sj T 2 P d for all triangles T 2 4g;
where P d is the space of polynomials of degree d,
