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Heisenberg’s and Hardy’s Uncertainty Principles for Special Relativistic Space-Time Fourier Transformation

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Abstract

The special relativistic (space-time) Fourier transform (SFT) in Clifford algebra \(Cl_{(3,1)}\) of space-time, first introduced from a mathematical point of view in Hitzer (Adv Appl Clifford Algebras 17:497–517, 2007), extends the quaternionic Fourier transform to functions, fields and signals in space-time. The purpose of this paper is to advance the study of the SFT and investigate important properties such as continuity, Plancherel identity, Riemann–Lebesgue lemma, and to establish the associated Hausdorff–Young inequality. Moreover, using the observer related space-time split several uncertainty inequalities are established, including Heisenberg’s uncertainty principle and Hardy’s theorem.

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Acknowledgements

The authors would like to express their sincere gratitude to the referees for their insightful and valuable comments and suggestions.

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Authors and Affiliations

  1. Ecole Normale Supérieure (ENS-Meknès), University Moulay Ismail, Meknes, 3104, Morocco

    Youssef El Haoui

  2. College of Liberal Arts, International Christian University, Osawa 3-10-2, Mitaka, Tokyo, 181-8585, Japan

    Eckhard Hitzer

  3. Equipe d’Analyse Harmonique et Probabilités, Department of Mathematics, Faculty of Sciences, University Moulay Ismail, Meknes, 11201, Morocco

    Youssef El Haoui & Said Fahlaoui

Authors
  1. Youssef El Haoui
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  2. Eckhard Hitzer
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  3. Said Fahlaoui
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Correspondence to Youssef El Haoui.

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Communicated by Uwe Kaehler

To Mrs. El Haoui.

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El Haoui, Y., Hitzer, E. & Fahlaoui, S. Heisenberg’s and Hardy’s Uncertainty Principles for Special Relativistic Space-Time Fourier Transformation. Adv. Appl. Clifford Algebras 30, 69 (2020). https://doi.org/10.1007/s00006-020-01093-5

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  • DOI: https://doi.org/10.1007/s00006-020-01093-5

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