WRONSKIANS OF FOURIER AND LAPLACE TRANSFORMS

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dc.contributor.authorDimitrov, Dimitar K. [UNESP]
dc.contributor.authorXu, Yuan
dc.contributor.institutionUniversidade Estadual Paulista (Unesp)
dc.contributor.institutionUniv Oregon
dc.date.accessioned2020-12-10T19:36:30Z
dc.date.available2020-12-10T19:36:30Z
dc.date.issued2019-09-15
dc.description.abstractAssociated with a given suitable function, or a measure, on R, we introduce a correlation function so that the Wronskian of the Fourier transform of the function is the Fourier transform of the corresponding correlation function, and the same holds for the Laplace transform. We obtain two types of results. First, we show that Wronskians of the Fourier transform of a non-negative function on R are positive definite functions and that the Wronskians of the Laplace transform of a nonnegative function on R+ are completely monotone functions. Then we establish necessary and sufficient conditions in order that a real entire function, defined as a Fourier transform of a positive kernel K, belongs to the Laguerre-Polya class, which answers an old question of Polya. The characterization is given in terms of a density property of the correlation kernel related to K, via classical results of Laguerre and Jensen and employing Wiener's L-1 Tauberian theorem. As a consequence, we provide a necessary and sufficient condition for the Riemann hypothesis in terms of a density of the translations of the correlation function related to the Riemann xi-function.en
dc.description.affiliationUniv Estadual Paulista, IBILCE, Dept Matemat Aplicada, BR-15054000 Sao Jose Do Rio Preto, SP, Brazil
dc.description.affiliationUniv Oregon, Dept Math, Eugene, OR 97403 USA
dc.description.affiliationUnespUniv Estadual Paulista, IBILCE, Dept Matemat Aplicada, BR-15054000 Sao Jose Do Rio Preto, SP, Brazil
dc.description.sponsorshipConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)
dc.description.sponsorshipFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)
dc.description.sponsorshipNSF
dc.description.sponsorshipIdCNPq: 306136/2017-1
dc.description.sponsorshipIdFAPESP: 2016/09906-0
dc.description.sponsorshipIdFAPESP: 2014/08328-8
dc.description.sponsorshipIdNSF: DMS-1510296
dc.format.extent4107-4125
dc.identifierhttp://dx.doi.org/10.1090/tran/7809
dc.identifier.citationTransactions Of The American Mathematical Society. Providence: Amer Mathematical Soc, v. 372, n. 6, p. 4107-4125, 2019.
dc.identifier.doi10.1090/tran/7809
dc.identifier.issn0002-9947
dc.identifier.urihttp://hdl.handle.net/11449/196191
dc.identifier.wosWOS:000487085100011
dc.language.isoeng
dc.publisherAmer Mathematical Soc
dc.relation.ispartofTransactions Of The American Mathematical Society
dc.sourceWeb of Science
dc.subjectFourier transform
dc.subjectLaplace transform
dc.subjectWronskian
dc.subjectentire function
dc.subjectLaguerre-Polya class
dc.subjectRiemann hypothesis
dc.titleWRONSKIANS OF FOURIER AND LAPLACE TRANSFORMSen
dc.typeArtigo
dcterms.rightsHolderAmer Mathematical Soc
unesp.author.orcid0000-0002-3078-2336[1]

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