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

TL;DR

This paper develops a multidimensional Euler--MacLaurin summation formula for polytopes and generalizes series expansions in Bernoulli polynomials, connecting Fourier analysis, polytope geometry, and multivariate Bernoulli polynomials.

Contribution

It introduces a new multidimensional summation formula on polytopes and extends Bernoulli polynomial expansions to a multivariate setting with applications to Fourier analysis.

Findings

Derived a weighted Euler--MacLaurin formula for polytopes

Generalized Mordell's series expansion using multivariate Bernoulli polynomials

Analyzed asymptotic behavior of Fourier transforms of characteristic functions

Abstract

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