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 $\mathfrak{f}$.

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.