Harmonic analysis on the Einstein gyrogroup
Autor(a) principal: | |
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Data de Publicação: | 2014 |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10773/14178 |
Resumo: | In this paper we study harmonic analysis on the Einstein gyrogroup of the open ball of R$^n$, $n \in N,$ centered at the origin and with arbitrary radius $t \in R^+,$ associated to the generalised Laplace-Beltrami operator $$ L_{\sigma,t} = \disp \left( 1 - \frac{\|x\|^2}{t^2} \right) \!\left( \Delta - \sum_{i,j=1}^n \frac{x_i x_j}{t^2} \frac{\partial^2}{\partial x_i \partial x_j} - \frac{\kappa}{t^2} \sum_{i=1}^n x_i \frac{\partial}{\partial x_i} + \frac{\kappa(2-\kappa)}{4t^2} \right)$$ where $\kappa=n+\sigma$ and $\sigma \in {\mathbb R}$ is an arbitrary parameter. The generalised harmonic analysis for $L_{\sigma,t}$ gives rise to the $(\sigma,t)$-translation, the $(\sigma,t)$-convolution, the $(\sigma,t)$-spherical Fourier transform, the $(\sigma,t)$-Poisson transform, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and Plancherel's Theorem. In the limit of large $t,$ $t \rightarrow +\infty,$ the resulting hyperbolic harmonic analysis tends to the standard Euclidean harmonic analysis on $R^n,$ thus unifying hyperbolic and Euclidean harmonic analysis. |
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Harmonic analysis on the Einstein gyrogroupEinstein gyrogroupGeneralised Helgason-Fourier transformSpherical functionsHyperbolic convolutionEigenfunctionsLaplace-Beltrami-operatorIn this paper we study harmonic analysis on the Einstein gyrogroup of the open ball of R$^n$, $n \in N,$ centered at the origin and with arbitrary radius $t \in R^+,$ associated to the generalised Laplace-Beltrami operator $$ L_{\sigma,t} = \disp \left( 1 - \frac{\|x\|^2}{t^2} \right) \!\left( \Delta - \sum_{i,j=1}^n \frac{x_i x_j}{t^2} \frac{\partial^2}{\partial x_i \partial x_j} - \frac{\kappa}{t^2} \sum_{i=1}^n x_i \frac{\partial}{\partial x_i} + \frac{\kappa(2-\kappa)}{4t^2} \right)$$ where $\kappa=n+\sigma$ and $\sigma \in {\mathbb R}$ is an arbitrary parameter. The generalised harmonic analysis for $L_{\sigma,t}$ gives rise to the $(\sigma,t)$-translation, the $(\sigma,t)$-convolution, the $(\sigma,t)$-spherical Fourier transform, the $(\sigma,t)$-Poisson transform, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and Plancherel's Theorem. In the limit of large $t,$ $t \rightarrow +\infty,$ the resulting hyperbolic harmonic analysis tends to the standard Euclidean harmonic analysis on $R^n,$ thus unifying hyperbolic and Euclidean harmonic analysis.JGSP2015-06-03T09:20:12Z2014-01-01T00:00:00Z2014info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10773/14178eng1312-519210.7546/jgsp-35-2014-21-60Ferreira, 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:RCAAP2024-02-22T11:25:52Zoai:ria.ua.pt:10773/14178Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T02:49:48.726181Repositório 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 Einstein gyrogroup |
title |
Harmonic analysis on the Einstein gyrogroup |
spellingShingle |
Harmonic analysis on the Einstein gyrogroup Ferreira, Milton Einstein gyrogroup Generalised Helgason-Fourier transform Spherical functions Hyperbolic convolution Eigenfunctions Laplace-Beltrami-operator |
title_short |
Harmonic analysis on the Einstein gyrogroup |
title_full |
Harmonic analysis on the Einstein gyrogroup |
title_fullStr |
Harmonic analysis on the Einstein gyrogroup |
title_full_unstemmed |
Harmonic analysis on the Einstein gyrogroup |
title_sort |
Harmonic analysis on the Einstein 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 |
Einstein gyrogroup Generalised Helgason-Fourier transform Spherical functions Hyperbolic convolution Eigenfunctions Laplace-Beltrami-operator |
topic |
Einstein gyrogroup Generalised Helgason-Fourier transform Spherical functions Hyperbolic convolution Eigenfunctions Laplace-Beltrami-operator |
description |
In this paper we study harmonic analysis on the Einstein gyrogroup of the open ball of R$^n$, $n \in N,$ centered at the origin and with arbitrary radius $t \in R^+,$ associated to the generalised Laplace-Beltrami operator $$ L_{\sigma,t} = \disp \left( 1 - \frac{\|x\|^2}{t^2} \right) \!\left( \Delta - \sum_{i,j=1}^n \frac{x_i x_j}{t^2} \frac{\partial^2}{\partial x_i \partial x_j} - \frac{\kappa}{t^2} \sum_{i=1}^n x_i \frac{\partial}{\partial x_i} + \frac{\kappa(2-\kappa)}{4t^2} \right)$$ where $\kappa=n+\sigma$ and $\sigma \in {\mathbb R}$ is an arbitrary parameter. The generalised harmonic analysis for $L_{\sigma,t}$ gives rise to the $(\sigma,t)$-translation, the $(\sigma,t)$-convolution, the $(\sigma,t)$-spherical Fourier transform, the $(\sigma,t)$-Poisson transform, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and Plancherel's Theorem. In the limit of large $t,$ $t \rightarrow +\infty,$ the resulting hyperbolic harmonic analysis tends to the standard Euclidean harmonic analysis on $R^n,$ thus unifying hyperbolic and Euclidean harmonic analysis. |
publishDate |
2014 |
dc.date.none.fl_str_mv |
2014-01-01T00:00:00Z 2014 2015-06-03T09:20:12Z |
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/10773/14178 |
url |
http://hdl.handle.net/10773/14178 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
1312-5192 10.7546/jgsp-35-2014-21-60 |
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 |
JGSP |
publisher.none.fl_str_mv |
JGSP |
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 |
instname_str |
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 |
repository.mail.fl_str_mv |
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1799137548829720576 |