Locally linearly independent bases for bivariate polynomial spline spaces

Locally linearly independent bases are constructed for the spaces S r d (4) of polynomial splines of degree d 3r + 2 and smoothness r deened on triangulations, as well as for their superspline subspaces. x1. Introduction Given a regular triangulation 4 of a set of vertices V, 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. Suppose B := fB i g n i=1 is a basis for S r d (4). Then B is said to be locally linearly independent (LLI) provided that for every T 2 4, the basis splines fB i g i2 T are linearly independent on T, where T := fi : T supp(B i)g: (1:1) Since S r d (4) contains the space P d of polynomials, B being LLI is equivalent to the condition ## T = dimP d = d + 2 2 for every T 2 4: (1:2) For a discussion of various equivalent deenitions of local linear independence, see 10,15]. Locally linearly independent bases play an important role in the theory of interpolation and almost interpolation by multivariate splines, see 14,15]. They are also of interest since an LLI basis B for S r d (4) is a least supported basis in the sense that it is optimal with respect to the size of the supports of the B i , see 7]. Locally supported bases have been constructed for the spline spaces S r d (4) and their superspline subspaces in 3,4,16,17,20,24], but they are mostly not LLI,