Some Results for the Windowed Fourier Transform Related to the Spherical Mean Operator
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Some Results for the Windowed Fourier Transform Related to the Spherical Mean Operator Khaled Hleili1,2 Received: 29 April 2019 / Revised: 11 February 2020 / Accepted: 2 March 2020 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract We define and study the windowed Fourier transform associated with the spherical mean operator. We prove the boundedness and compactness of localization operators of this windowed. Next, we establish new uncertainty principles for the Fourier and the windowed Fourier transforms associated with the spherical mean operator. More precisely, we give a Shapiro-type uncertainty inequality for the Fourier transform that is, for s > 0 and {αk }k be an orthonormal sequence in L2 (dνn+1 ) M
s |(r, x)|s αk 22,νn+1 + |(μ, λ)|s F˜ (αk )22,νn+1 Rn,s M 1+ 2n+1 .
k=1
Finally, we prove an analogous inequality for the windowed Fourier transform. Keywords Spherical mean operator · Windowed fourier transform · Localization operators · Shapiro’s uncertainty principles Mathematics Subject Classification (2010) 42B10 · 42C25
1 Introduction Time frequency analysis [6] plays an important role in harmonic analysis, in particular in signal theory. In this context and motivated by quantum mechanics, the physicist Dennis Gabor [4] has introduced the Gabor transform, in which he uses translation, convolution, and modulation operators of a single Gaussian to represent one-dimensional signal. In [18], Shapiro has studied the localization for an orthonormal sequence (φk )k∈N . He showed that if the means and the dispersions of the orthonormal sequence (φk )k∈N and their Khaled Hleili
[email protected] 1
Department of Mathematics, Preparatory Institute for Engineering Studies of Kairouan, Kairouan, Tunisia
2
Department of Mathematics, Faculty of Science, Northern Borders University, Arar, Saudi Arabia
K. Hleili
Fourier transforms (φk )k∈N are uniformly bounded, then (φk )k∈N is finite. In [9], the authors gave a quantitative version of the precedent theorem, that is if (φk )k∈N is an orthonormal sequence in L2 (R), then for every n ∈ N n k=1
(n + 1)2 . xφk 22 + y φk 22 2π
The spherical mean operator R is defined, for a function f on R × Rn , even with respect to the first variable [14], by R (f )(r, x) = f (rη, x + rξ )dσn (η, ξ ), (r, x) ∈ R × Rn , Sn
where S n is the unit sphere of R × Rn and dσn is the surface measure on S n normalized to have total measure one. The operator R has many important physical applications, namely in image processing of so-called synthetic aperture radar (SAR) data [7, 8, 17, 21], or in the linearized inverse scattering problem in acoustics [3]. The Fourier transform F associated with the spherical mean operator is defined for every integrable function f on [0, +∞[×Rn with respect to the measure dνn+1 , by +∞ f (r, x)R cos(s.)e−iy|. (r, x)dνn+1 (r, x), ∀(s, y) ∈ Υ, F (f )(s, y) = 0
Rn
where dνn+1 is the measure defined on [0, +∞[×Rn by dνn+1 (r, x) = 2
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