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 realvalued 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 socalled 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 deletioncontraction decomposition.
 Publication:

arXiv eprints
 Pub Date:
 November 2012
 DOI:
 10.48550/arXiv.1211.1187
 arXiv:
 arXiv:1211.1187
 Bibcode:
 2012arXiv1211.1187L
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Commutative Algebra;
 Mathematics  Numerical Analysis;
 Primary: 05B35;
 41A05;
 41A15;
 52B20;
 Secondary: 13F20;
 41A63;
 47F05;
 52B40;
 52C07
 EPrint:
 10 pages, 3 figures