Polynomial-reproducing spline spaces from fine zonotopal tilings

TL;DR

This paper establishes a connection between polynomial-reproducing spline spaces and fine zonotopal tilings, providing a general framework that includes cases with dependent points and offering algorithms for practical spline construction.

Contribution

It introduces a new link between spline spaces and zonotopal tilings, generalizes known results, and develops algorithms for spline construction and evaluation.

Findings

Established a connection between spline spaces and zonotopal tilings.

Proved the existence of an iterative construction process for these spline spaces.

Developed practical algorithms for constructing and evaluating spline functions.

Abstract

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.