多面体复形上的格支样条
Lattice-Supported Splines on Polytopal Complexes
摘要：
通过分析由$\mathcal{P}$的内余维一个面的交格得到的某些子复形$\mathcal{P}_W$，研究了纯$n$维多面体复形$\mathcal{P}\subset\mathbb{R}^n$上的分段光滑多项式函数$r$的模$C^r(\mathcal{P})$.我们得到两个主要结果：首先，我们证明了在足够高的次数下，次数为$\leq k$的样条的向量空间$C^r_k(\mathcal{P})$具有由支持在$k\gg0$的$\mathcal{P}_W$上的样条构成的基。我们称这种样条为格支样条。这表明，一个类似的概念，一个明星支持的基础$C^r_k(Δ)$研究的阿尔伯克基Schumaker在单纯的情况下举行。第二，我们提供了一对图解，一个涉及格子支撑样条，界定多大$k$必须使$\mbox{dim}_\mathbb{R} C^r_k(\mathcal{P})$同意麦当劳-申克给出的公式。一族例子表明后一个猜想是紧的。所提出的界限推广了单纯形情况下的已知和已知界限。
英文摘要：
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 $C^r_k(\mathcal{P})$ of splines of degree $\leq k$ has a basis consisting of splines supported on the $\mathcal{P}_W$ for $k\gg0$. We call such splines lattice-supported. This shows that an analog of the notion of a star-supported basis for $C^r_k(Δ)$ 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.
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