Matthias Lenz - Interpolation, box splines, and lattice points in zonotopes

dmtcs:12820 - Discrete Mathematics & Theoretical Computer Science, January 1, 2013, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) - https://doi.org/10.46298/dmtcs.12820
Interpolation, box splines, and lattice points in zonotopesArticle

Authors: Matthias Lenz 1

  • 1 Mathematical Institute [Oxford]

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.


Volume: DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)
Section: Proceedings
Published on: January 1, 2013
Imported on: November 21, 2016
Keywords: matroid,zonotope,lattice points,interpolation,box spline,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]

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