Homology of smooth splines: generic triangulations and a conjecture of Strang

Abstract:

For $\Delta$ a triangulated $d$-dimensional region in ${{\mathbf {R}}^d}$, let $S_m^r(\Delta )$ denote the vector space of all ${C^r}$ functions $F$ on $\Delta$ that, restricted to any simplex in $\Delta$, are given by polynomials of degree at most $m$. We consider the problem of computing the dimension of such spaces. We develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds $\Delta$ in the plane, getting lower bounds on the dimension of $S{}_m^r(\Delta )$ for all $r$. For $r = 1$, we prove a conjecture of Strang concerning the generic dimension of the space of ${C^1}$ splines over a triangulated manifold in ${{\mathbf {R}}^2}$. Finally, we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region.