Euler-MacLaurin summation formula on polytopes and expansions in multivariate Bernoulli polynomials

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 $q\chi _{\tau\mathcal{P}}$.