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Orientador(es)
Resumo(s)
In 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.
Descrição
Keywords (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.
Article number: 70.
Acknowledgements: 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.
Article number: 70.
Acknowledgements: 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.
Palavras-chave
Janssen’s representation Quaternion hyperbolic frame operator Quaternion hyperbolic windowed Fourier transform Quaternionic hyperbolic Gabor frames Quaternionic hyperbolic Poisson summation formula Walnut’s representation Wexler–Raz biorthogonality
Contexto Educativo
Citação
Ferreira, 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
Editora
Springer Nature