Some results on the lattice parameters of quaternionic Gabor frames
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/10773/14587 |
Resumo: | Gabor frames play a vital role not only modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this work we present a non-trivial generalization of Gabor frames to the quaternionic case and give new density results. The key tool is the two-sided windowed quaternionic Fourier transform (WQFT). As in the complex case, we want to write the WQFT as an inner product between a quaternion-valued signal and shifts and modulates of a real-valued window function. We demonstrate a Heisenberg uncertainty principle and for the results regarding the density, we employ the quaternionic Zak transform to obtain necessary and sufficient conditions to ensure that a quaternionic Gabor system is a quaternionic Gabor frame. We conclude with a proof that the Gabor conjecture do not hold true in the quaternionic case. |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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Some results on the lattice parameters of quaternionic Gabor framesQuaternionic Gabor systemQuaternionic Gabor framesWindowed quaternionic Fourier transformQuaternionic Zak transformCritical densityGabor frames play a vital role not only modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this work we present a non-trivial generalization of Gabor frames to the quaternionic case and give new density results. The key tool is the two-sided windowed quaternionic Fourier transform (WQFT). As in the complex case, we want to write the WQFT as an inner product between a quaternion-valued signal and shifts and modulates of a real-valued window function. We demonstrate a Heisenberg uncertainty principle and for the results regarding the density, we employ the quaternionic Zak transform to obtain necessary and sufficient conditions to ensure that a quaternionic Gabor system is a quaternionic Gabor frame. We conclude with a proof that the Gabor conjecture do not hold true in the quaternionic case.Springer2015-09-01T13:54:10Z2015-01-01T00:00:00Z2015info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfapplication/pdfhttp://hdl.handle.net/10773/14587eng0188-700910.1007/s00006-015-0587-0Hartmann, Stefaninfo: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:RCAAP2024-02-22T11:26:20Zoai:ria.ua.pt:10773/14587Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T02:50:01.038856Repositório 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 |
Some results on the lattice parameters of quaternionic Gabor frames |
title |
Some results on the lattice parameters of quaternionic Gabor frames |
spellingShingle |
Some results on the lattice parameters of quaternionic Gabor frames Hartmann, Stefan Quaternionic Gabor system Quaternionic Gabor frames Windowed quaternionic Fourier transform Quaternionic Zak transform Critical density |
title_short |
Some results on the lattice parameters of quaternionic Gabor frames |
title_full |
Some results on the lattice parameters of quaternionic Gabor frames |
title_fullStr |
Some results on the lattice parameters of quaternionic Gabor frames |
title_full_unstemmed |
Some results on the lattice parameters of quaternionic Gabor frames |
title_sort |
Some results on the lattice parameters of quaternionic Gabor frames |
author |
Hartmann, Stefan |
author_facet |
Hartmann, Stefan |
author_role |
author |
dc.contributor.author.fl_str_mv |
Hartmann, Stefan |
dc.subject.por.fl_str_mv |
Quaternionic Gabor system Quaternionic Gabor frames Windowed quaternionic Fourier transform Quaternionic Zak transform Critical density |
topic |
Quaternionic Gabor system Quaternionic Gabor frames Windowed quaternionic Fourier transform Quaternionic Zak transform Critical density |
description |
Gabor frames play a vital role not only modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this work we present a non-trivial generalization of Gabor frames to the quaternionic case and give new density results. The key tool is the two-sided windowed quaternionic Fourier transform (WQFT). As in the complex case, we want to write the WQFT as an inner product between a quaternion-valued signal and shifts and modulates of a real-valued window function. We demonstrate a Heisenberg uncertainty principle and for the results regarding the density, we employ the quaternionic Zak transform to obtain necessary and sufficient conditions to ensure that a quaternionic Gabor system is a quaternionic Gabor frame. We conclude with a proof that the Gabor conjecture do not hold true in the quaternionic case. |
publishDate |
2015 |
dc.date.none.fl_str_mv |
2015-09-01T13:54:10Z 2015-01-01T00:00:00Z 2015 |
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/10773/14587 |
url |
http://hdl.handle.net/10773/14587 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
0188-7009 10.1007/s00006-015-0587-0 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf 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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1799137550825160704 |