Heisenberg uncertainty principle for Gabor transform on compact extensions of Rn

We prove in this paper a generalization of Heisenberg inequality for Gabor transform in the setup of the semidirect product R n ⋊ K , where K is a compact subgroup of automorphisms of R n . We also solve the sharpness problem and thus we obtain an optimal analogue of the Heisenberg inequality. A loc...

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Bibliographic Details
Published in:Journal of pseudo-differential operators and applications 2024-06, Vol.15 (2)
Main Authors: Smaoui, Kais, Abid, Khouloud
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Automorphisms
Functional Analysis
Gabor transformation
Mathematics
Mathematics and Statistics
Operator Theory
Partial Differential Equations
Subgroups
Uncertainty principles
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Summary:We prove in this paper a generalization of Heisenberg inequality for Gabor transform in the setup of the semidirect product R n ⋊ K , where K is a compact subgroup of automorphisms of 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.
ISSN:1662-9981
1662-999X
DOI:10.1007/s11868-024-00598-y