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

• Published in: Annali dell'Università di Ferrara. Sezione 7. Scienze matematiche Vol. 65; no. 1; pp. 139 - 170
• Main Authors: Rachdi, Lakhdar Tannech, Amri, Besma, Hammami, Aymen
• Format: Journal Article
• Language: English
• Published: Milan Springer Milan 01.05.2019; Springer Nature B.V
• Subjects: Algebraic Geometry; Analysis; Differential equations; Fourier transforms; Gabor transformation; Geometry; History of Mathematical Sciences; Mathematics; Mathematics and Statistics; Numerical Analysis; Operators (mathematics); Time-frequency analysis; Uncertainty principles; Inversion formula; Uncertainty principle; 44A35; Local uncertainty principle; Plancherel theorem; Time frequency; 42A38; Windowed Fourier transform; Annihilating subset
• ISSN: 0430-3202; 1827-1510
• DOI: 10.1007/s11565-018-0311-9

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 , + ∞ [ × 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 , + ∞ [ × 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.