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Let  X  be a list of vectors that is unimodular and let  B   X   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 ) B   X   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.

Lattice Points in Polytopes, Box Splines, and Todd Operators