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Gabor frames in quaternionic analysis: a transition from Euclidean to hyperbolic structures

2025-09-13Artigo científico Acesso aberto
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datacite.subject.fosCiências Naturais::Matemáticas
datacite.subject.sdg04:Educação de Qualidade
dc.contributor.authorFerreira, M.
dc.contributor.authorHartmann, S.
dc.contributor.authorMorais, J.
dc.date.accessioned2026-10-20T15:49:28Z
dc.date.available2026-10-20T15:49:28Z
dc.date.issued2025-09-13
dc.descriptionKeywords (post print): Quaternion hyperbolic windowed Fourier transform, Quaternionic hyperbolic Poisson summation formula, Quaternionic hyperbolic Gabor frames, Quaternion hyperbolic frame operator Walnut's representation, Janssen's representation, Wexler-Raz biorthogonality. 2000 MSC: Primary 42C15, 30G35; Secondary 42B10, 51M10, 47S05.
dc.descriptionArticle number: 70.
dc.descriptionAcknowledgements: The work of M. Ferreira was supported by CIDMA under the FCT (Portuguese Foundation for Science and Technology) Multi-Annual Financing Program for R&D Units. The third author's work was supported by the Asociación Mexicana de Cultura, A. C.
dc.description.abstractIn this paper, we introduce a novel integral transform termed the two-sided Qua\-ternion Hyperbolic Windowed Fourier Transform (ts-QHWFT), designed for the analysis of two-dimen\-sional quaternion-valued signals defined in an open rectangle of the plane equipped with a hyperbolic measure. We investigate the basic properties of the ts-QHWFT, including modulation, hyperbolic translation, hyperbolic dilation, Moyal's identity, and an inversion formula, which are essential for our analysis. The proposed ts-QHWFT enables the construction of quaternionic hyperbolic Gabor frames employing innovative time-frequency analysis tools such as the relativistic version of the Wiener space of quaternion-valued signals, the hyperbolic Poisson summation formula, and the correlation function. We then establish hyperbolic analogs of Walnut's and Janssen's representations of the quaternionic Gabor frame, as well as quaternionic versions of the Wexler-Raz biorthogonality and Ron-Shen duality. We also provide a detailed characterization of quaternionic hyperbolic tight Gabor frames. This algebraic approach provides a hyperbolic counterpart of the corresponding results in the quaternionic Euclidean scenario. Finally, as an application, we demonstrate the utility of the proposed ts-QHWFT for the analysis of two-dimensional linear space-varying hyperbolic systems, illustrating the results of this work.eng
dc.description.sponsorshipThe work of M. Ferreira was supported by CIDMA under the FCT (Portuguese Foundation for Science and Technology) Multi-Annual Financing Program for R&D Units. The third author's work was supported by the Asociación Mexicana de Cultura, A. C.
dc.identifier.citationFerreira, M., Hartmann, S. & Morais, J. Gabor frames in quaternionic analysis: a transition from Euclidean to hyperbolic structures. Banach J. Math. Anal. 19(4), 70 (2025). https://doi.org/10.1007/s43037-025-00453-2
dc.identifier.doi10.1007/s43037-025-00453-2
dc.identifier.eissn2662-2033
dc.identifier.issn1735-8787
dc.identifier.urihttp://hdl.handle.net/10400.8/17091
dc.language.isoeng
dc.peerreviewedyes
dc.publisherSpringer Nature
dc.relation.hasversionhttps://link.springer.com/article/10.1007/s43037-025-00453-2
dc.relation.ispartofBanach Journal of Mathematical Analysis
dc.rights.urihttp://creativecommons.org/licenses/by/4.0/
dc.subjectJanssen’s representation
dc.subjectQuaternion hyperbolic frame operator
dc.subjectQuaternion hyperbolic windowed Fourier transform
dc.subjectQuaternionic hyperbolic Gabor frames
dc.subjectQuaternionic hyperbolic Poisson summation formula
dc.subjectWalnut’s representation
dc.subjectWexler–Raz biorthogonality
dc.titleGabor frames in quaternionic analysis: a transition from Euclidean to hyperbolic structureseng
dc.typejournal article
dspace.entity.typePublication
oaire.citation.endPage30
oaire.citation.issue4
oaire.citation.startPage1
oaire.citation.titleBanach Journal of Mathematical Analysis
oaire.citation.volume19
oaire.versionhttp://purl.org/coar/version/c_ab4af688f83e57aa
person.familyNameFerreira
person.familyNameMorais
person.givenNameMilton
person.givenNameJoão
person.identifier.ciencia-idCA19-2009-F26D
person.identifier.ciencia-id3614-652A-118E
person.identifier.orcid0000-0003-1816-8293
person.identifier.orcid0000-0003-3406-2878
person.identifier.ridA-2004-2015
person.identifier.scopus-author-id12144179800
relation.isAuthorOfPublicationb1460cdc-4ced-46c6-a637-68b425d104dc
relation.isAuthorOfPublication3bc0f910-a460-461f-863a-1ca701ee597f
relation.isAuthorOfPublication.latestForDiscoveryb1460cdc-4ced-46c6-a637-68b425d104dc

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