Harmonic analysis on the proper velocity gyrogroup
Autor(a) principal: | |
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Data de Publicação: | 2017 |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10773/16613 |
Resumo: | In this paper we study harmonic analysis on the Proper Velocity (PV) gyrogroup using the gyrolanguage of analytic hyperbolic geometry. PV addition is the relativistic addition of proper velocities in special relativity and it is related with the hyperboloid model of hyperbolic geometry. The generalized harmonic analysis depends on a complex parameter $z$ and on the radius $t$ of the hyperboloid and comprises the study of the generalized translation operator, the associated convolution operator, the generalized Laplace-Beltrami operator and its eigenfunctions, the generalized Poisson transform and its inverse, the generalized Helgason-Fourier transform, its inverse and Plancherel's Theorem. In the limit of large $t,$ $t \rightarrow +\infty,$ the generalized harmonic analysis on the hyperboloid tends to the standard Euclidean harmonic analysis on ${\mathbb R}^n,$ thus unifying hyperbolic and Euclidean harmonic analysis. |
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Harmonic analysis on the proper velocity gyrogroupPV gyrogroupLaplace Beltrami operatorEigenfunctionsGeneralized Helgason-Fourier transformPlancherel's TheoremIn this paper we study harmonic analysis on the Proper Velocity (PV) gyrogroup using the gyrolanguage of analytic hyperbolic geometry. PV addition is the relativistic addition of proper velocities in special relativity and it is related with the hyperboloid model of hyperbolic geometry. The generalized harmonic analysis depends on a complex parameter $z$ and on the radius $t$ of the hyperboloid and comprises the study of the generalized translation operator, the associated convolution operator, the generalized Laplace-Beltrami operator and its eigenfunctions, the generalized Poisson transform and its inverse, the generalized Helgason-Fourier transform, its inverse and Plancherel's Theorem. In the limit of large $t,$ $t \rightarrow +\infty,$ the generalized harmonic analysis on the hyperboloid tends to the standard Euclidean harmonic analysis on ${\mathbb R}^n,$ thus unifying hyperbolic and Euclidean harmonic analysis.Duke University Press2017-01-09T18:44:31Z2017-01-01T00:00:00Z2017-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10773/16613eng1735-8787Ferreira, Miltoninfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2024-02-22T11:31:06Zoai:ria.ua.pt:10773/16613Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T02:51:44.308482Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
Harmonic analysis on the proper velocity gyrogroup |
title |
Harmonic analysis on the proper velocity gyrogroup |
spellingShingle |
Harmonic analysis on the proper velocity gyrogroup Ferreira, Milton PV gyrogroup Laplace Beltrami operator Eigenfunctions Generalized Helgason-Fourier transform Plancherel's Theorem |
title_short |
Harmonic analysis on the proper velocity gyrogroup |
title_full |
Harmonic analysis on the proper velocity gyrogroup |
title_fullStr |
Harmonic analysis on the proper velocity gyrogroup |
title_full_unstemmed |
Harmonic analysis on the proper velocity gyrogroup |
title_sort |
Harmonic analysis on the proper velocity gyrogroup |
author |
Ferreira, Milton |
author_facet |
Ferreira, Milton |
author_role |
author |
dc.contributor.author.fl_str_mv |
Ferreira, Milton |
dc.subject.por.fl_str_mv |
PV gyrogroup Laplace Beltrami operator Eigenfunctions Generalized Helgason-Fourier transform Plancherel's Theorem |
topic |
PV gyrogroup Laplace Beltrami operator Eigenfunctions Generalized Helgason-Fourier transform Plancherel's Theorem |
description |
In this paper we study harmonic analysis on the Proper Velocity (PV) gyrogroup using the gyrolanguage of analytic hyperbolic geometry. PV addition is the relativistic addition of proper velocities in special relativity and it is related with the hyperboloid model of hyperbolic geometry. The generalized harmonic analysis depends on a complex parameter $z$ and on the radius $t$ of the hyperboloid and comprises the study of the generalized translation operator, the associated convolution operator, the generalized Laplace-Beltrami operator and its eigenfunctions, the generalized Poisson transform and its inverse, the generalized Helgason-Fourier transform, its inverse and Plancherel's Theorem. In the limit of large $t,$ $t \rightarrow +\infty,$ the generalized harmonic analysis on the hyperboloid tends to the standard Euclidean harmonic analysis on ${\mathbb R}^n,$ thus unifying hyperbolic and Euclidean harmonic analysis. |
publishDate |
2017 |
dc.date.none.fl_str_mv |
2017-01-09T18:44:31Z 2017-01-01T00:00:00Z 2017-01 |
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/10773/16613 |
url |
http://hdl.handle.net/10773/16613 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
1735-8787 |
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 |
Duke University Press |
publisher.none.fl_str_mv |
Duke University Press |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
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Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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1799137566819090432 |