Local asymptotic Euler-Maclaurin expansion for Riemann sums over a semi-rational polyhedron

2015/01/24 by Nicole Berline, Berline, Nicole, Michèle Vergne +1 · 1 citation
Mathematics · #05A15 #52B20 #65B15 #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Mathematical and Theoretical Analysis #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1502.01671

openalex publication_date 2015/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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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