Positive-definiteness and integral representations for special functions

Bibliographic Details
Main Author: Buescu, Jorge
Publication Date: 2020
Other Authors: Paixão, António
Format: Article
Language: eng
Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
Download full: http://hdl.handle.net/10400.21/12608
Summary: It is known that a holomorphic positive definite function f defined on a horizontal strip of the complex plane may be characterized as the Fourier-Laplace transform of a unique exponentially finite measure on R. In this paper we apply this complex integral representation to specific families of special functions, including the Gamma, zeta and Bessel functions, and construct explicitly the corresponding measures, thus providing new insight into the nature of complex positive and co-positive definite functions. In the case of the zeta function this process leads to a new proof of an integral representation on the critical strip.
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spelling Positive-definiteness and integral representations for special functionsPositive definite functionsFourier-Laplace transformCharacteristic functionsHolomorphyGamma functionZeta functionExponentially convex functionsIt is known that a holomorphic positive definite function f defined on a horizontal strip of the complex plane may be characterized as the Fourier-Laplace transform of a unique exponentially finite measure on R. In this paper we apply this complex integral representation to specific families of special functions, including the Gamma, zeta and Bessel functions, and construct explicitly the corresponding measures, thus providing new insight into the nature of complex positive and co-positive definite functions. In the case of the zeta function this process leads to a new proof of an integral representation on the critical strip.SpringerRCIPLBuescu, JorgePaixão, António2021-01-14T12:45:42Z2020-082020-08-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10400.21/12608eng1385-12921572-928110.1007/s11117-020-00784-4info:eu-repo/semantics/openAccessreponame:Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)instname:FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologiainstacron:RCAAP2025-02-12T08:28:32Zoai:repositorio.ipl.pt:10400.21/12608Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireinfo@rcaap.ptopendoar:https://opendoar.ac.uk/repository/71602025-05-28T19:55:57.652776Repositórios Científicos de Acesso Aberto de Portugal (RCAAP) - FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologiafalse
dc.title.none.fl_str_mv Positive-definiteness and integral representations for special functions
title Positive-definiteness and integral representations for special functions
spellingShingle Positive-definiteness and integral representations for special functions
Buescu, Jorge
Positive definite functions
Fourier-Laplace transform
Characteristic functions
Holomorphy
Gamma function
Zeta function
Exponentially convex functions
title_short Positive-definiteness and integral representations for special functions
title_full Positive-definiteness and integral representations for special functions
title_fullStr Positive-definiteness and integral representations for special functions
title_full_unstemmed Positive-definiteness and integral representations for special functions
title_sort Positive-definiteness and integral representations for special functions
author Buescu, Jorge
author_facet Buescu, Jorge
Paixão, António
author_role author
author2 Paixão, António
author2_role author
dc.contributor.none.fl_str_mv RCIPL
dc.contributor.author.fl_str_mv Buescu, Jorge
Paixão, António
dc.subject.por.fl_str_mv Positive definite functions
Fourier-Laplace transform
Characteristic functions
Holomorphy
Gamma function
Zeta function
Exponentially convex functions
topic Positive definite functions
Fourier-Laplace transform
Characteristic functions
Holomorphy
Gamma function
Zeta function
Exponentially convex functions
description It is known that a holomorphic positive definite function f defined on a horizontal strip of the complex plane may be characterized as the Fourier-Laplace transform of a unique exponentially finite measure on R. In this paper we apply this complex integral representation to specific families of special functions, including the Gamma, zeta and Bessel functions, and construct explicitly the corresponding measures, thus providing new insight into the nature of complex positive and co-positive definite functions. In the case of the zeta function this process leads to a new proof of an integral representation on the critical strip.
publishDate 2020
dc.date.none.fl_str_mv 2020-08
2020-08-01T00:00:00Z
2021-01-14T12:45:42Z
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://hdl.handle.net/10400.21/12608
url http://hdl.handle.net/10400.21/12608
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv 1385-1292
1572-9281
10.1007/s11117-020-00784-4
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
instname:FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologia
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reponame_str Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
collection Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
repository.name.fl_str_mv Repositórios Científicos de Acesso Aberto de Portugal (RCAAP) - FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologia
repository.mail.fl_str_mv info@rcaap.pt
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