Interpolation, box splines, and lattice points in zonotopes

TL;DR

This paper proves a conjecture that functions on lattice points inside zonotopes can be uniquely extended using box splines and differential operators, revealing connections to matroid theory.

Contribution

It establishes the unique extension of functions on lattice points in zonotopes via box splines and differential operators, confirming a conjecture by Holtz and Ron.

Findings

Functions on lattice points can be extended using box splines.

The extension is unique and involves internal P-space differential operators.

Connections to matroid theory and deletion-contraction are identified.

Abstract

Let $X$ be a totally unimodular list of vectors in some lattice. Let $B_X$ be the box spline defined by $X$. Its support is the zonotope $Z(X)$. We show that any real-valued function defined on the set of lattice points in the interior of $Z(X)$ can be extended to a function on $Z(X)$ of the form $p(D)B_X$ in a unique way, where $p(D)$ is a differential operator that is contained in the so-called internal $\Pcal$-space. This was conjectured by Olga Holtz and Amos Ron. We also point out connections between this interpolation problem and matroid theory, including a deletion-contraction decomposition.