Harmonic analysis on the Möbius gyrogroup
Autor(a) principal: | |
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Data de Publicação: | 2015 |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10773/14179 |
Resumo: | In this paper we propose to develop harmonic analysis on the Poincaré ball $B_t^n$, a model of the n-dimensional real hyperbolic space. The Poincaré ball $B_t^n$ is the open ball of the Euclidean n-space $R^n$ with radius $t>0$, centered at the origin of $R^n$ and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in $\mathbb{R}^n$. For any $t>0$ and an arbitrary parameter $\sigma \in R$ we study the $(\sigma,t)$-translation, the $( \sigma,t)$-convolution, the eigenfunctions of the $(\sigma,t)$-Laplace-Beltrami operator, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and the associated Plancherel's Theorem, which represent counterparts of standard tools, thus, enabling an effective theory of hyperbolic harmonic analysis. Moreover, when $t \rightarrow +\infty$ the resulting hyperbolic harmonic analysis on $B_t^n$ tends to the standard Euclidean harmonic analysis on $R^n$, thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on $B_t^n$. |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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7160 |
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Harmonic analysis on the Möbius gyrogroupMöbius gyrogroupHelgason-Fourier transformSpherical functionsHyperbolic convolutionEigenfunctions of the Laplace-Beltrami-operatorDiffusive waveletsIn this paper we propose to develop harmonic analysis on the Poincaré ball $B_t^n$, a model of the n-dimensional real hyperbolic space. The Poincaré ball $B_t^n$ is the open ball of the Euclidean n-space $R^n$ with radius $t>0$, centered at the origin of $R^n$ and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in $\mathbb{R}^n$. For any $t>0$ and an arbitrary parameter $\sigma \in R$ we study the $(\sigma,t)$-translation, the $( \sigma,t)$-convolution, the eigenfunctions of the $(\sigma,t)$-Laplace-Beltrami operator, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and the associated Plancherel's Theorem, which represent counterparts of standard tools, thus, enabling an effective theory of hyperbolic harmonic analysis. Moreover, when $t \rightarrow +\infty$ the resulting hyperbolic harmonic analysis on $B_t^n$ tends to the standard Euclidean harmonic analysis on $R^n$, thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on $B_t^n$.Springer2015-042015-04-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10773/14179eng1531-585110.1007/s00041-014-9370-1Ferreira, 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:RCAAP2024-02-22T11:25:51Zoai:ria.ua.pt:10773/14179Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T02:49:48.412551Repositório 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 Möbius gyrogroup |
title |
Harmonic analysis on the Möbius gyrogroup |
spellingShingle |
Harmonic analysis on the Möbius gyrogroup Ferreira, Milton Möbius gyrogroup Helgason-Fourier transform Spherical functions Hyperbolic convolution Eigenfunctions of the Laplace-Beltrami-operator Diffusive wavelets |
title_short |
Harmonic analysis on the Möbius gyrogroup |
title_full |
Harmonic analysis on the Möbius gyrogroup |
title_fullStr |
Harmonic analysis on the Möbius gyrogroup |
title_full_unstemmed |
Harmonic analysis on the Möbius gyrogroup |
title_sort |
Harmonic analysis on the Möbius 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 |
Möbius gyrogroup Helgason-Fourier transform Spherical functions Hyperbolic convolution Eigenfunctions of the Laplace-Beltrami-operator Diffusive wavelets |
topic |
Möbius gyrogroup Helgason-Fourier transform Spherical functions Hyperbolic convolution Eigenfunctions of the Laplace-Beltrami-operator Diffusive wavelets |
description |
In this paper we propose to develop harmonic analysis on the Poincaré ball $B_t^n$, a model of the n-dimensional real hyperbolic space. The Poincaré ball $B_t^n$ is the open ball of the Euclidean n-space $R^n$ with radius $t>0$, centered at the origin of $R^n$ and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in $\mathbb{R}^n$. For any $t>0$ and an arbitrary parameter $\sigma \in R$ we study the $(\sigma,t)$-translation, the $( \sigma,t)$-convolution, the eigenfunctions of the $(\sigma,t)$-Laplace-Beltrami operator, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and the associated Plancherel's Theorem, which represent counterparts of standard tools, thus, enabling an effective theory of hyperbolic harmonic analysis. Moreover, when $t \rightarrow +\infty$ the resulting hyperbolic harmonic analysis on $B_t^n$ tends to the standard Euclidean harmonic analysis on $R^n$, thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on $B_t^n$. |
publishDate |
2015 |
dc.date.none.fl_str_mv |
2015-04 2015-04-01T00:00:00Z |
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/10773/14179 |
url |
http://hdl.handle.net/10773/14179 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
1531-5851 10.1007/s00041-014-9370-1 |
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: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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1799137548815040512 |