Local Euler-Maclaurin formula for polytopes
نویسندگان
چکیده
(with DF = 1 for F = P). As explained in [2], essential properties required on the operators DF are ”locality” and ”computability”. At each face F of P, the operator DF should depend only of the translation class modulo Z of the normal cone No(P, F ) to P at a generic point of F (if F is of codimension k, the normal cone is an affine cone of dimension k). In particular if P is an integral polytope, the operator DF should depend only of the cone of normal feasible directions at F to P.
منابع مشابه
Asymptotic Euler-Maclaurin formula for Delzant polytopes
Formulas for the Riemann sums over lattice polytopes determined by the lattice points in the polytopes are often called Euler-Maclaurin formulas. An asymptotic Euler-Maclaurin formula, by which we mean an asymptotic expansion formula for Riemann sums over lattice polytopes, was first obtained by Guillemin-Sternberg [GS]. Then, the problem is to find a concrete formula for the each term of the e...
متن کاملSum-integral Interpolators and the Euler-maclaurin Formula for Polytopes
A local lattice point counting formula, and more generally a local Euler-Maclaurin formula follow by comparing two natural families of meromorphic functions on the dual of a rational vector space V , namely the family of exponential sums (S) and the family of exponential integrals (I) parametrized by the set of rational polytopes in V . The paper introduces the notion of an interpolator between...
متن کاملExact Euler Maclaurin Formulas for Simple Lattice Polytopes
Euler Maclaurin formulas for a polytope express the sum of the values of a function over the lattice points in the polytope in terms of integrals of the function and its derivatives over faces of the polytope or its dilations. There are two kinds of Euler Maclaurin formulas: exact formulas, which apply to exponential or polynomial functions, and formulas with remainder, which apply to arbitrary...
متن کاملThe Euler-Maclaurin formula for simple integral polytopes.
We give a Euler-Maclaurin formula with remainder for the sum of a smooth function on the integral points in a simple integral lattice polytope. Our proof uses elementary methods.
متن کاملLattice Points in Simple Polytopes
P (h) φ(x)dx where the polytope P (h) is obtained from P by independent parallel motions of all facets. This extends to simple lattice polytopes the EulerMaclaurin summation formula of Khovanskii and Pukhlikov [8] (valid for lattice polytopes such that the primitive vectors on edges through each vertex of P form a basis of the lattice). As a corollary, we recover results of Pommersheim [9] and ...
متن کامل