Harmonic analysis problems associated with the k -Hankel Gabor transform
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Harmonic analysis problems associated with the k-Hankel Gabor transform Hatem Mejjaoli1 · Salem Ben Saïd2 Received: 24 February 2020 / Revised: 3 July 2020 / Accepted: 7 July 2020 © Springer Nature Switzerland AG 2020
Abstract We introduce a continuous k-Hankel Gabor transform acting on a Hilbert space deforming L 2 (R). We prove a Plancherel formula and L 2 -inversion formulas for it. We also prove several uncertainty principles for this transform such as Heisenberg type inequalities and Faris–Price type uncertainty principle. Keywords k-Hankel transform · k-Hankel Gabor transform · Plancherel formula · Inversion theorem · Heisenberg’s type inequality · Local Cowling–Price’s type inequalities · Faris–Price’s uncertainty principle Mathematics Subject Classification Primary 26D10 · 43A32; Secondary 33C52 · 43A15 · 44A15
1 Introduction In quantum mechanics, the Heisenberg uncertainty principle states that the position and momentum of a particle described by a wave function in L 2 (R) cannot be simultaneously and arbitrary small. Motivated by this principle in 1946, D. Gabor, who won the 1971 Nobel Prize in physics, first recognized the great importance of localized time and frequency concentrations in signal processing [24]. In order to incorporate both
Dedicated to the spirit of Professor Ahmed Fitouhi.
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Hatem Mejjaoli [email protected] Salem Ben Saïd [email protected]
1
College of Sciences, Department of Mathematics, Taibah University, PO BOX 30002, Al Madinah AL Munawarah, Saudi Arabia
2
Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain, Abu Dhabi, UAE
H. Mejjaoli and S. B. Saïd
time and frequency localization properties in one single transform function, Gabor introduced the windowed Fourier transform (or Gabor transform) by using a Gaussian distribution function as window function. Subsequently, various other functions have been used as window functions instead of the Gaussian function that was originally introduced by Gabor. The Gabor transformation has been found to be very useful in many physical and engineering applications, including wave propagation, signal processing and quantum optics [11,12]. For more details on the Gabor transform and its basic properties, we refer the reader to [18]. We may also refer to [25] where the author extends Gabor theory to the setup of locally compact abelian groups, and to [46] for the Gabor transform on Gelfand pairs. We note also that the notion of the Gabor transform for strong hypergroups was first introduced by Czaja and Gigante [9]. Motivated by the previous works, in this paper we extend the Gabor transform to the setup of the minimal unitary representation of the conformal group O(n + 1, 1), and then we prove its fundamental properties. More precisely, in [2] the authors gave a far reaching extension of the classical Fourier analysis by constructing a generalized Fourier transform Fk,a acting on a Hilbert space deforming L 2 (Rn ). The deformation parameters consists of a real parameter a >
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