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Consider the Riemann sum of a smooth compactly supported function h(x) on apolyhedron in R^d, sampled at the points of the lattice Z^d/t. We give anasymptotic expansion when t goes to infinity, writing each coefficient of thisexpansion as a sum indexed by the faces f of the polyhedron, where the f-termis 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 thedifferential operator for the face f can be chosen (in a unique way) to involveonly normal derivatives to f. Our formulas are valid for a semi-rationalpolyhedron and a real sampling parameter t, if we allow for step-polynomialcoefficients, instead of just constant ones.

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