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Uncertainty principles and time frequency analysis related to the Riemann–Liouville operator

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Abstract

We define and study the windowed Fourier transform, called also the Gabor transform, associated with singular partial differential operators defined on the half plane \(]0,+\infty [\times \mathbb {R}\). We prove a Plancherel theorem and an inversion formula that we use to establish the classical Heisenberg uncertainty principle. Next, we study this transform on subsets of \(([0,+\infty [\times \mathbb {R})^2\) with finite measures, in particular we establish a well generalized Heisenberg–Pauli–Weyl uncertainty principle for this transform (with general magnitude). Also, we check a local uncertainty principle and we give nice applications.

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Authors and Affiliations

  1. UR11ES23 Analyse géométrique et harmonique, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092, Tunis, Tunisia

    Lakhdar Tannech Rachdi, Besma Amri & Aymen Hammami

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  1. Lakhdar Tannech Rachdi
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  2. Besma Amri
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  3. Aymen Hammami
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Correspondence to Lakhdar Tannech Rachdi.

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Rachdi, L.T., Amri, B. & Hammami, A. Uncertainty principles and time frequency analysis related to the Riemann–Liouville operator. Ann Univ Ferrara 65, 139–170 (2019). https://doi.org/10.1007/s11565-018-0311-9

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  • DOI: https://doi.org/10.1007/s11565-018-0311-9

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