Year: 2014
Type: article
Abstract: Let X be a list of vectors that is unimodular and let BX be the box spline defined by X. In a previous article, the author proved that every real-valued function on the set of interior lattice points of the zonotope defined by X can be extended to a function on the whole zonotope of the form p(D)BX in a unique way, where p(D) is a differential operator that is contained in the so-called internal |$\mathcal {P}$|-space. In this paper, we construct an explicit solution to this interpolation problem as a truncation of the Todd operator. This slightly generalizes the De Concini–Procesi–Vergne deconvolution formula. As a corollary we obtain a slight generalization of the Khovanskii–Pukhlikov formula that relates the volume and the number of integer points in a smooth lattice polytope.
DOI: 10.1093/imrn/rnu095
Author: Matthias Lenz
Institution: University of Oxford
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2022084010