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

TL;DR

This paper develops an asymptotic expansion for Riemann sums over semi-rational polyhedra, expressing coefficients as face integrals involving differential operators, with applications to lattice point enumeration.

Contribution

It introduces a local asymptotic Euler-Maclaurin expansion for Riemann sums over semi-rational polyhedra, including explicit formulas involving normal derivatives and step-polynomial coefficients.

Findings

Asymptotic expansion valid for large t

Coefficients expressed as face integrals with differential operators

Operators involve only normal derivatives when a Euclidean scalar product is chosen

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.