Given a point configuration A, we uncover a connection between polynomial-reproducing spline spaces over subsets of conv(A) and fine zonotopal tilings of the zonotope Z(V) associated to the corresponding vector configuration. This link directly generalizes a known result on Delaunay configurations and naturally encompasses, due to its combinatorial character, the case of repeated and affinely dependent points in A. We prove the existence of a general iterative construction process for such spaces. Finally, we turn our attention to regular fine zonotopal tilings, specializing our previous results and exploiting the adjacency graph of the tiling to propose a set of practical algorithms for the construction and evaluation of the associated spline functions.

翻译：根据一个点配置 A,我们发现在相对应矢量配置相关联的锥形(A)子集上多成再生样板空间与Z(V)子集上Z(V)的微子圆形砖块之间有联系。这种联系直接概括了Delaunay配置的已知结果,并自然包括了A中重复的和紧密依附的点的情况。我们证明存在此类空间的一般迭接构建过程。最后,我们把注意力转向常规微小的圆顶层砖块,专门研究我们以前的结果,并利用该顶层的相邻图来提出一套实用的算法,用于构建和评估相关的螺纹函数。