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 χτP denotes the characteristic function of a dilated integer convex polytope P and q is a function with suitable regularity, we prove that the periodization of qχτP 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 qχτP^ , the Fourier transform of qχτP .

Brandolini, L., Colzani, L., Gariboldi, B., Gigante, G., Monguzzi, A. (2023). Euler–MacLaurin Summation Formula on Polytopes and Expansions in Multivariate Bernoulli Polynomials. JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, 29(3), 1-49 [10.1007/s00041-023-10011-9].

Euler–MacLaurin Summation Formula on Polytopes and Expansions in Multivariate Bernoulli Polynomials

Colzani, L.;Gariboldi, B.;
2023

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 χτP denotes the characteristic function of a dilated integer convex polytope P and q is a function with suitable regularity, we prove that the periodization of qχτP 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 qχτP^ , the Fourier transform of qχτP .
Articolo in rivista - Articolo scientifico
Bernoulli polynomials; Euler–MacLaurin summation formula; Fourier transform;
English
30-mag-2023
2023
29
3
1
49
33
open
Brandolini, L., Colzani, L., Gariboldi, B., Gigante, G., Monguzzi, A. (2023). Euler–MacLaurin Summation Formula on Polytopes and Expansions in Multivariate Bernoulli Polynomials. JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS, 29(3), 1-49 [10.1007/s00041-023-10011-9].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/418678
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