多面体上的欧拉-麦克劳林求和公式与多元伯努利多项式的展开式
Euler-MacLaurin summation formula on polytopes and expansions in multivariate Bernoulli polynomials
摘要：
本文给出了多面体上的一个多维加权欧拉-麦克劳林求和公式，并推广了L.关于伯努利多项式的级数展开。这些结果是后果的一个更一般的系列扩展，即，如果$χ_{τ\mathcal{P}}$表示的特征函数的扩张整数凸多面体$\mathcal{P}$和$q$是一个功能，具有适当的规律性，我们证明了$qχ_{τ\mathcal{P}}$的周期化承认多元伯努利多项式的扩展。这些多元多项式与Lerch Zeta函数有关。为了证明我们的结果，我们需要仔细研究$\widehat{qχ_{τ\mathcal{P}}}$的渐近展开，$qχ_{τ\mathcal{P}}$的傅立叶变换。
英文摘要：
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 $χ_{τ\mathcal{P}}$ 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 $qχ_{τ\mathcal{P}}$ 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 $\widehat{qχ_{τ\mathcal{P}}}$, the Fourier transform of $qχ_{τ\mathcal{P}}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
