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<p>In this paper, we introduce a novel integral transform termed the two-sided Quaternion Hyperbolic Windowed Fourier Transform (ts-QHWFT), designed for the analysis of two-dimensional 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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年9月13日

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  • M. Ferreira,
  • S. Hartmann,
  • J. Morais

摘要

In this paper, we introduce a novel integral transform termed the two-sided Quaternion Hyperbolic Windowed Fourier Transform (ts-QHWFT), designed for the analysis of two-dimensional 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.