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

Abstract

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