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Quaternion Hyperbolic Fourier Transforms and Uncertainty Principles

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Abstract

The present study introduces the two-sided and right-sided Quaternion Hyperbolic Fourier Transforms (QHFTs) for analyzing two-dimensional quaternion-valued signals defined in an open rectangle of the Euclidean plane endowed with a hyperbolic measure. The different forms of these transforms are defined by replacing the Euclidean plane waves with the corresponding hyperbolic plane waves in one dimension, giving the hyperbolic counterpart of the corresponding Euclidean Quaternion Fourier Transforms. Using hyperbolic geometry tools, we study the main operational and mapping properties of the QHFTs, such as linearity, shift, modulation, dilation, symmetry, inversion, and derivatives. Emphasis is placed on novel hyperbolic derivative and hyperbolic primitive concepts, which lead to the differentiation and integration properties of the QHFTs. We further prove the Riemann–Lebesgue Lemma and Parseval’s identity for the two-sided QHFT. Besides, we establish the Logarithmic, Heisenberg–Weyl, Donoho–Stark, and Benedicks’ uncertainty principles associated with the two-sided QHFT by invoking hyperbolic counterparts of the convolution, Pitt’s inequality, and the Poisson summation formula. This work is motivated by the potential applications of the QHFTs and the analysis of the corresponding hyperbolic quaternionic signals.

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Acknowledgements

The work of M. Ferreira was supported by Portuguese funds through CIDMA-Center for Research and Development in Mathematics and Applications, and FCT—Fundação para a Ciência e a Tecnologia, within projects with DOI https://doi.org/10.54499/UIDB/04106/2020 and https://doi.org/10.54499/UIDP/04106/2020. The second author’s work was supported by the Asociación Mexicana de Cultura, A. C.

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Authors and Affiliations

  1. School of Technology and Management, Polytechnic of Leiria, 2411-901, Leiria, Portugal

    M. Ferreira

  2. CIDMA - Center for Research and Development in Mathematics and Applications, University of Aveiro, Campus Universitário de Santiago, 3810-193, Aveiro, Portugal

    M. Ferreira

  3. Department of Mathematics, ITAM, Río Hondo #1, Col. Progreso Tizapán, 01080, Mexico City, Mexico

    J. Morais

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M.F. and J.M. have discussed the results and contributed to the final manuscript.

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This article is part of the topical collection “Higher Dimensional Geometric Function Theory and Hypercomplex Analysis” edited by Irene Sabadini, and Daniele Struppa.

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Ferreira, M., Morais, J. Quaternion Hyperbolic Fourier Transforms and Uncertainty Principles. Complex Anal. Oper. Theory 18, 16 (2024). https://doi.org/10.1007/s11785-023-01451-8

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