Interpolation, box splines, and lattice points in zonotopes

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.