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Register a new userLocal Euler-Maclaurin formula for polytopes
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Authors
Nicole Berline
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We give a local Euler-Maclaurin formula for rational convex polytopes in a rational euclidean space . For every affine rational polyhedral cone C in a rational euclidean space W, we construct a differential operator of infinite order D(C) on W with constant rational coefficients, which is unchanged when C is translated by an integral vector. Then for every convex rational polytope P in a rational euclidean space V and every polynomial function f (x) on V, the sum of the values of f(x) at the integral points of P is equal to the sum, for all faces F of P, of the integral over F of the function D(N(F)).f, where we denote by N(F) the normal cone to P along F.
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Consider the Riemann sum of a smooth compactly supported function h(x) on a polyhedron in R^d, sampled at the points of the lattice Z^d/t. We give an asymptotic expansion when t goes to infinity, writing each coefficient of this expansion as a sum indexed by the faces f of the polyhedron, where the f-term is the integral over f of a differential operator applied to the function h(x). In particular, if a Euclidean scalar product is chosen, we prove that the differential operator for the face f can be chosen (in a unique way) to involve only normal derivatives to f. Our formulas are valid for a semi-rational polyhedron and a real sampling parameter t, if we allow for step-polynomial coefficients, instead of just constant ones.
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