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Novel Gabor-Type Transform and Weighted Uncertainty Principles

Author

Listed:
  • Saifallah Ghobber

    (Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia)

  • Hatem Mejjaoli

    (Department of Mathematics, College of Sciences, Taibah University, P.O. Box 30002, Al Madinah AL Munawarah 42353, Saudi Arabia)

Abstract

The linear canonical Fourier transform is one of the most celebrated time-frequency tools for analyzing non-transient signals. In this paper, we will introduce and study the deformed Gabor transform associated with the linear canonical Dunkl transform (LCDT). Then, we will formulate several weighted uncertainty principles for the resulting integral transform, called the linear canonical Dunkl-Gabor transform (LCDGT). More precisely, we will prove some variations in Heisenberg’s uncertainty inequality. Then, we will show an analog of Pitt’s inequality for the LCDGT and formulate a Beckner-type uncertainty inequality via two approaches. Finally, we will derive a Benedicks-type uncertainty principle for the LCDGT, which shows the impossibility of a non-trivial function and its LCDGT to both be supported on sets of finite measure. As a side result, we will prove local uncertainty principles for the LCDGT.

Suggested Citation

  • Saifallah Ghobber & Hatem Mejjaoli, 2025. "Novel Gabor-Type Transform and Weighted Uncertainty Principles," Mathematics, MDPI, vol. 13(7), pages 1-37, March.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:7:p:1109-:d:1622287
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    References listed on IDEAS

    as
    1. Jun-Fang Zhang & Shou-Ping Hou, 2012. "The Generalization of the Poisson Sum Formula Associated with the Linear Canonical Transform," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-9, December.
    2. Navneet Kaur & Bivek Gupta & Amit K. Verma & Ravi P. Agarwal, 2024. "Offset Linear Canonical Stockwell Transform for Boehmians," Mathematics, MDPI, vol. 12(15), pages 1-18, July.
    3. Didar Urynbassarova & Aajaz A. Teali, 2023. "Convolution, Correlation, and Uncertainty Principles for the Quaternion Offset Linear Canonical Transform," Mathematics, MDPI, vol. 11(9), pages 1-24, May.
    4. Hehe Yang & Qiang Feng & Xiaoxia Wang & Didar Urynbassarova & Aajaz A. Teali, 2024. "Reduced Biquaternion Windowed Linear Canonical Transform: Properties and Applications," Mathematics, MDPI, vol. 12(5), pages 1-21, March.
    5. Jun-Fang Zhang & Shou-Ping Hou, 2012. "The Generalization of the Poisson Sum Formula Associated with the Linear Canonical Transform," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    Full references (including those not matched with items on IDEAS)

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    1. Hehe Yang & Qiang Feng & Xiaoxia Wang & Didar Urynbassarova & Aajaz A. Teali, 2024. "Reduced Biquaternion Windowed Linear Canonical Transform: Properties and Applications," Mathematics, MDPI, vol. 12(5), pages 1-21, March.

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