Multi-variable sinc integrals and volumes of polyhedra
type: Item
instance of: scholarly article
title: Multi-variable sinc integrals and volumes of polyhedra
zbMATH Open document ID: 1012.42005
DOI: 10.1023/A:1015727317007
author: David Borwein
author: Jonathan M. Borwein
author: Bernard A. jun. Mares
published in: The Ramanujan Journal
publication date: 2002-08-21T00:00:00Z
review text: \textit{D. Borwein} and \textit{J. M. Borwein} [Ramanujan J. 5, No. 1, 73-89 (2001; Zbl 0991.42004)] obtained interesting results for evaluating the integrals \[ I_n:=\int_0^\infty \prod_{j=1}^{1+n} \text{ sinc}(a_jx) dx, \] where \(\text{sinc}(x):={\sin x\over x}\). In this paper the authors investigate the multi-variable sinc integrals \[ \sigma(S):=\int_{{\mathbb R}^m} \prod_{k=1}^{m+n} \text{ sinc}(s_ky) dy \] where \(S=(s_1,\cdots,s_{m+n}), \;s_k\in {\mathbb R}^m.\) They obtain the results concerning the relationship between the integral \(\sigma(S)\) and the volume of the associated symmetric convex polyhedron. Also they establish an explicit algebraic formula for the computation of \(\sigma(S)\) which generalizes the result obtained by Borwein and Borwein in the one dimensional case (\(m=1\)) in the paper mentioned above .
reviewed by: Wang Kunyang
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 51M25
zbMATH DE Number: 1785614
zbMATH Keywords: multi-variable sinc integrals
zbMATH Keywords: Fourier transform
zbMATH Keywords: convolution
zbMATH Keywords: polyhedra
MaRDI profile type: Publication
Wikidata QID: Q56935778
description: scientific article; zbMATH DE number 1785614