WRONSKIANS OF FOURIER AND LAPLACE TRANSFORMS
Autor(a) principal: | |
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Data de Publicação: | 2019 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UNESP |
Texto Completo: | http://dx.doi.org/10.1090/tran/7809 http://hdl.handle.net/11449/196191 |
Resumo: | Associated 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. |
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WRONSKIANS OF FOURIER AND LAPLACE TRANSFORMSFourier transformLaplace transformWronskianentire functionLaguerre-Polya classRiemann hypothesisAssociated 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.Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)NSFUniv Estadual Paulista, IBILCE, Dept Matemat Aplicada, BR-15054000 Sao Jose Do Rio Preto, SP, BrazilUniv Oregon, Dept Math, Eugene, OR 97403 USAUniv Estadual Paulista, IBILCE, Dept Matemat Aplicada, BR-15054000 Sao Jose Do Rio Preto, SP, BrazilCNPq: 306136/2017-1FAPESP: 2016/09906-0FAPESP: 2014/08328-8NSF: DMS-1510296Amer Mathematical SocUniversidade Estadual Paulista (Unesp)Univ OregonDimitrov, Dimitar K. [UNESP]Xu, Yuan2020-12-10T19:36:30Z2020-12-10T19:36:30Z2019-09-15info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article4107-4125http://dx.doi.org/10.1090/tran/7809Transactions Of The American Mathematical Society. Providence: Amer Mathematical Soc, v. 372, n. 6, p. 4107-4125, 2019.0002-9947http://hdl.handle.net/11449/19619110.1090/tran/7809WOS:000487085100011Web of Sciencereponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengTransactions Of The American Mathematical Societyinfo:eu-repo/semantics/openAccess2021-10-23T04:53:26Zoai:repositorio.unesp.br:11449/196191Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462021-10-23T04:53:26Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false |
dc.title.none.fl_str_mv |
WRONSKIANS OF FOURIER AND LAPLACE TRANSFORMS |
title |
WRONSKIANS OF FOURIER AND LAPLACE TRANSFORMS |
spellingShingle |
WRONSKIANS OF FOURIER AND LAPLACE TRANSFORMS Dimitrov, Dimitar K. [UNESP] Fourier transform Laplace transform Wronskian entire function Laguerre-Polya class Riemann hypothesis |
title_short |
WRONSKIANS OF FOURIER AND LAPLACE TRANSFORMS |
title_full |
WRONSKIANS OF FOURIER AND LAPLACE TRANSFORMS |
title_fullStr |
WRONSKIANS OF FOURIER AND LAPLACE TRANSFORMS |
title_full_unstemmed |
WRONSKIANS OF FOURIER AND LAPLACE TRANSFORMS |
title_sort |
WRONSKIANS OF FOURIER AND LAPLACE TRANSFORMS |
author |
Dimitrov, Dimitar K. [UNESP] |
author_facet |
Dimitrov, Dimitar K. [UNESP] Xu, Yuan |
author_role |
author |
author2 |
Xu, Yuan |
author2_role |
author |
dc.contributor.none.fl_str_mv |
Universidade Estadual Paulista (Unesp) Univ Oregon |
dc.contributor.author.fl_str_mv |
Dimitrov, Dimitar K. [UNESP] Xu, Yuan |
dc.subject.por.fl_str_mv |
Fourier transform Laplace transform Wronskian entire function Laguerre-Polya class Riemann hypothesis |
topic |
Fourier transform Laplace transform Wronskian entire function Laguerre-Polya class Riemann hypothesis |
description |
Associated 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. |
publishDate |
2019 |
dc.date.none.fl_str_mv |
2019-09-15 2020-12-10T19:36:30Z 2020-12-10T19:36:30Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://dx.doi.org/10.1090/tran/7809 Transactions Of The American Mathematical Society. Providence: Amer Mathematical Soc, v. 372, n. 6, p. 4107-4125, 2019. 0002-9947 http://hdl.handle.net/11449/196191 10.1090/tran/7809 WOS:000487085100011 |
url |
http://dx.doi.org/10.1090/tran/7809 http://hdl.handle.net/11449/196191 |
identifier_str_mv |
Transactions Of The American Mathematical Society. Providence: Amer Mathematical Soc, v. 372, n. 6, p. 4107-4125, 2019. 0002-9947 10.1090/tran/7809 WOS:000487085100011 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Transactions Of The American Mathematical Society |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
4107-4125 |
dc.publisher.none.fl_str_mv |
Amer Mathematical Soc |
publisher.none.fl_str_mv |
Amer Mathematical Soc |
dc.source.none.fl_str_mv |
Web of Science reponame:Repositório Institucional da UNESP instname:Universidade Estadual Paulista (UNESP) instacron:UNESP |
instname_str |
Universidade Estadual Paulista (UNESP) |
instacron_str |
UNESP |
institution |
UNESP |
reponame_str |
Repositório Institucional da UNESP |
collection |
Repositório Institucional da UNESP |
repository.name.fl_str_mv |
Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP) |
repository.mail.fl_str_mv |
|
_version_ |
1799965364225835008 |