Lattice-Supported Splines on Polytopal Complexes

Abstract

We study the module $C^r(\mathcal{P})$ of piecewise polynomial functions of smoothness $r$ on a pure $n$-dimensional polytopal complex $\mathcal{P}\subset {\mathbb{R}}^n$, via an analysis of certain subcomplexes ${\mathcal{P}}_W$ obtained from the intersection lattice of the interior codimension one faces of $\mathcal{P}$. We obtain two main results: first, we show that in sufficiently high degree, the vector space $C_k^r(\mathcal{P})$ of splines of degree $\le k$ has a basis consisting of splines supported on the ${\mathcal{P}}_W$ for $k\gg 0$. We call such splines lattice-supported. This shows that an analog of the notion of a star-supported basis for $C_k^r(\Delta )$ studied by Alfeld-Schumaker in the simplicial case holds. Second, we provide a pair of conjectures, one involving lattice-supported splines, bounding how large $k$ must be so that $\mbox{dim}_\mathbb{R} C^r_k(\mathcal{P})$ agrees with the formula given by McDonald-Schenck. A family of examples shows that the latter conjecture is tight. The proposed bounds generalize known and conjectured bounds in the simplicial case.

22 pages, 11 figures. v2 (updated from published version): More examples added, as well as new results for the graded case. Index of summation in Definition 4.1 changed, theorems and proofs updated to reflect this

Cited by in corpus (2)

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• Associated Primes of Spline Complexes