  We provide a multidimensional weighted Euler--MacLaurin summation formula on
polytopes and a multidimensional generalization of a result due to L. J.
Mordell on the series expansion in Bernoulli polynomials. These results are
consequences of a more general series expansion; namely, if $\chi
_{\tau\mathcal{P}}$ denotes the characteristic function of a dilated integer
convex polytope $\mathcal{P}$ and $q$ is a function with suitable regularity,
we prove that the periodization of $q\chi_{\tau\mathcal{P}}$ admits an
expansion in terms of multivariate Bernoulli polynomials. These multivariate
polynomials are related to the Lerch Zeta function. In order to prove our
results we need to carefully study the asymptotic expansion of
$\widehat{q\chi_{\tau\mathcal{P}}}$, the Fourier transform of $q\chi
_{\tau\mathcal{P}}$.
