Interpolation, box splines, and lattice points in zonotopes
Abstrak
Given a finite list of vectors $X \subseteq \mathbb{R}^d$, one can define the box spline $B_X$. Box splines are piecewise polynomial functions that are used in approximation theory. They are also interesting from a combinatorial point of view and many of their properties solely depend on the structure of the matroid defined by the list $X$. The support of the box spline is the zonotope $Z(X)$. We show that if the list $X$ is totally unimodular, 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 $\mathcal{P}$-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.
Topik & Kata Kunci
Penulis (1)
Matthias Lenz
Akses Cepat
- Tahun Terbit
- 2013
- Sumber Database
- DOAJ
- DOI
- 10.46298/dmtcs.12820
- Akses
- Open Access ✓