Quantitative Uncertainty Principles Associated with the Deformed Gabor Transform

• Published in: Journal of mathematics (Hidawi) 2022-01, Vol.2022 (1)
• Main Authors: Mejjaoli, Hatem, Shah, Firdous A.
• Format: Article
• Language: English
• Subjects: Deformation; Gabor transformation; Harmonic analysis; Inequalities; Mathematics; Principles; Uncertainty principles
• ISSN: 2314-4629; 2314-4785
• DOI: 10.1155/2022/8658689

The highlights of the article are pointed out as follows: (1) To present the notion of generalized Gabor transform operator in the setting of the deformed Gabor transform (2) To obtain some Heisenberg-type uncertainty principles by following diverse strategies, such as the principle of nonexclusive entropy, the contraction semigroup method, and so on (3) Formulation of Pitt’s and Beckner’s uncertainty principles associated with DGT (4) To study some weighted uncertainty inequalities pertaining to the DGT (5) To obtain few other uncertainty inequalities for the deformed Gabor transform based on concentration over sets, such as the Benedick–Amrein–Berthier and the local-type uncertainty principles The remainder of this paper is organized into four sections: [...]in Section 5, we obtain some other uncertainty inequalities for the DGT based on concentration over sets, such as the Benedick–Amrein–Berthier and the local-type uncertainty principles. 2. [...]in case fx=Fx∈Lk,n1ℝ and is also even over ℝ, then (15)ℱk,nfξ=n/22 nk−n/2Γ2nk+2−n/2∫0∞Frj2nk−n/2nrξ1/nr2/n2nk+2−n/2−1dr, ∀ξ∈ℝ. For all f in Cb,eℝ, we have (31)τxk,nfy=Mk,n2n∫0πF 《x,y》ϕ,n1+−1nn!sgnxy2kn−nnCnnk−n/2cos ϕsin ϕ2nk−ndϕ . Besides the aforementioned trigonometric form, the authors in [14, 15] have also studied certain important properties of the generalized translation operator, which play a crucial role in the subsequent developments on the subject.