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  • 1 Tunis El Manar University, Faculty of Sciences of Tunis, Department of Mathematics, CAMPUS, 2092 Tunis, Tunisia
  • 2 Taibah University, College of Sciences, Department of Mathematics, PO BOX 30002 Al Madinah AL Munawarah, Saudi Arabia
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We introduce the directional short-time Fourier transform for which we prove a new Plancherel’s formula. We also prove for this transform several uncertainty principles as Heisenberg inequalities, logarithmic uncertainty principle, Faris–Price uncertainty principles and Donoho–Stark’s uncertainty principles.

  • [1]

    Atanasova, S., Pilipovi’c, S. and Saneva, K., Directional time-frequency analysis and directional regularity, Bull. Malays. Math. Sci. Soc., 42 (2019), 20752090.

    • Search Google Scholar
    • Export Citation
  • [2]

    Beckner, W., Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc., 123 (1995), 18971905.

  • [3]

    Bialynicki-Birula, I., Entropic uncertainty relations in quantum mechanics, in Quantum probability and applications II, pages 90–103. Springer, 1985.

    • Search Google Scholar
    • Export Citation
  • [4]

    Bonami, A. Demange, B. and Jaming, P., Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms, Revista Matemàtica Iberoamericana, 19(1) (2003), 2355.

    • Search Google Scholar
    • Export Citation
  • [5]

    Candès, E. J., Harmonic Analysis of Neural Networks, Appl. Comput. Harmon. Anal., 6(2) (1999), 197218.

  • [6]

    Candès, E. J. and Donoho, D. L., New tight frames of curvelets and optimal representations of objects with piesewise-C 2 singularities, Comm. Pure Appl. Math., 57 (2004), 219266.

    • Search Google Scholar
    • Export Citation
  • [7]

    Cook, C. E, and Bernfeld, M., Radar Signals An Introduction to Theory and Applications, Academic Press, New York, 1967.

  • [8]

    Donoho, D. L. and Stark, P. B., Uncertainty principles and signal recovery, SIAM Journal on Applied Mathematics, 49(3) (1989), 906931.

    • Search Google Scholar
    • Export Citation
  • [9]

    Faris, W. G., Inequalities and uncertainty principles, J. Math. Phys., 19 (1978), 461466.

  • [10]

    Folland, G-B. and Sitaram, A., The uncertainty principle: a Mathematical survey, Journal of Fourier analysis and applications, 3(3) (1997), 207238.

    • Search Google Scholar
    • Export Citation
  • [11]

    Gabor, D., Theory of communication. Part 1: The analysis of information, Journal of the Institution of Electrical Engineers-Part III: Radio and Communication Engineering, 93(26) (1946), 429441.

    • Search Google Scholar
    • Export Citation
  • [12]

    Grafakos, L. and Sansing, C., Gabor frames and directional time–frequency analysis, Applied and Computational Harmonic Analysis, 25(1) (2008), 4767.

    • Search Google Scholar
    • Export Citation
  • [13]

    Gröchenig, K., Foundations of time-frequency analysis, Springer Science & Business Media, 2013.

  • [14]

    Havin, V. and Jöricke, B., The uncertainty principle in harmonic analysis, volume 24, Berlin: Springer Verlag, 1994.

  • [15]

    Heisenberg, W., Über den anschaulichen Inhalt der quantentheoretischen Kinematic und Mechanik, Zeitschrift fÃijr Physik, 43 (1927),172âĂŞ-198; The Physical Principles of the Quantum Theory (Dover, New York, 1949; The Univ. Chicago Press, 1930).

    • Search Google Scholar
    • Export Citation
  • [16]

    Helgason, S., The Radon Transform, Progress in Mathematics, Birkhäuser Boston, 1999.

  • [17]

    Hosseini Giv, H., Directional short-time Fourier transform, Journal of Mathematical Analysis and Applications, 399(1) (2013), 100107.

    • Search Google Scholar
    • Export Citation
  • [18]

    Mejjaoli, H. and Omri, S., Spectral theorems associated with the directional short-time Fourier transform, Journal of Pseudo-Differential Operators and Applications, 11 (2020), 1554.

    • Search Google Scholar
    • Export Citation
  • [19]

    Price, J. F., Inequalities and local uncertainty principles, Math. Phys., 24 (1978), 17111714.

  • [20]

    Price, J. F., Sharp local uncertainty principles, Studia Math., 85, (1987), 3745.

  • [21]

    Price, J. F. and Sitaram, A., Local uncertainty inequalities for locally compact groups, Trans. Amer. Math. Soc., 308 (1988), 105114.

    • Search Google Scholar
    • Export Citation
  • [22]

    Ricaud, B. and Torrésani, B., A survey of uncertainty principles and some signal processing applications, Adv. Comput. Math., 40(3) (2014), 629650.

    • Search Google Scholar
    • Export Citation
  • [23]

    Saneva, K. H.-V. and Atanasova, S., Directional short-time Fourier transform of distributions, Journal of Inequalities and Applications, 2016(124) (2016).

    • Search Google Scholar
    • Export Citation
  • [24]

    Shannon, C-E., A mathematical theory of communication, ACM SIGMOBILE Mobile Computing and Communications Review, 5(1) (2001), 355.

  • [25]

    Woodward, P. M., Probability and information theory, with applicetions to radar, Pergamon Press, Oxford, second edition, 1964.