Local Asymptotic Euler-Maclaurin Expansion for Riemann Sums over a Semi-Rational Polyhedron

Abstract

Consider the Riemann sum of a smooth compactly supported function h(x) on a polyhedron \(\mathfrak{p} \subseteq \mathbb{R}^{d}\), sampled at the points of the lattice \(\mathbb{Z}^{d}/t\). We give an asymptotic expansion when \(t \rightarrow +\infty\), writing each coefficient of this expansion as a sum indexed by the faces \(\mathfrak{f}\) of the polyhedron, where the \(\mathfrak{f}\) term is the integral over \(\mathfrak{f}\) 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 \(\mathfrak{f}\) can be chosen (in a unique way) to involve only normal derivatives to \(\mathfrak{f}\).