A new bound for smooth spline spaces

Abstract

For a planar simplicial complex $\Delta \subseteq {\mathbb{R}}^2$, Schumaker proves in [22] that a lower bound on the dimension of the space $C_k^r(\Delta )$ of planar splines of smoothness $r$ and degree $k$ on $\Delta$ is given by a polynomial $P_{\Delta }(r,k)$, and Alfeld–Schumaker show in [2] that $P_{\Delta }(r,k)$ gives the correct dimension when $k\ge 4r+1$. Examples due to Morgan–Scott, Tohaneanu, and Yuan show that the equality $C_k^r=P_{\Delta }(r,k)$ can fail for $k\in \{2r,2r+1\}$. In this note we prove that the equality $C_k^r=P_{\Delta }(r,k)$ cannot hold in general for $k\le (22r+7)/10$.