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Heisenberg uncertainty principle for Gabor transform on compact extensions of \(\mathbb {R}^n\)

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Abstract

We prove in this paper a generalization of Heisenberg inequality for Gabor transform in the setup of the semidirect product \(\mathbb {R}^n\rtimes K\), where K is a compact subgroup of automorphisms of \(\mathbb {R}^n\). We also solve the sharpness problem and thus we obtain an optimal analogue of the Heisenberg inequality. A local uncertainty inequality for the Gabor transform is also provided, in the same context. This allows us to prove a couple of global uncertainty inequalities. The representation theory and Plancherel formula are fundamental tools in the proof of our results.

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Authors and Affiliations

  1. Higher Institute of Information Technology and Multimedia, Pôle Technologique de Sfax, Sfax University, Route de Tunis 242, Sfax, Tunisia

    Kais Smaoui

  2. Faculty of Sciences, Sfax University, Route de Soukra, 3038, Sfax, Tunisia

    Khouloud Abid

Authors
  1. Kais Smaoui
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  2. Khouloud Abid
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All authors reviewed the manuscript.

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Correspondence to Kais Smaoui.

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Smaoui, K., Abid, K. Heisenberg uncertainty principle for Gabor transform on compact extensions of \(\mathbb {R}^n\). J. Pseudo-Differ. Oper. Appl. 15, 28 (2024). https://doi.org/10.1007/s11868-024-00598-y

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  • DOI: https://doi.org/10.1007/s11868-024-00598-y

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