A remark on the convolution with the box spline
نویسنده
چکیده
The semi-discrete convolution with the box spline is an important tool in approximation theory. We give a formula for the difference between semidiscrete convolution and convolution with the box spline. This formula involves multiple Bernoulli polynomials. 1. Box splines and semi-discrete convolution Let V be a n-dimensional real vector space equipped with a lattice Λ. If we choose a basis of the lattice Λ, then we may identify V with Rn and Λ with Zn. We choose here the Lebesgue measure dv associated to the lattice Λ. Let X = [a1, a2, . . . , aN ] be a sequence (a multiset) of N nonzero vectors in Λ. The zonotope Z(X) associated with X is the polytope
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