Uncertainty Principles for the Generalized Fourier Transform Associated with Spherical Mean Operator

2013 ◽  
Vol 29 (4) ◽  
pp. 309-332
Author(s):  
H. Mejjaoli and Y. Othmani
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principles ◽  
Generalized Fourier Transform ◽  
Spherical Mean Operator ◽  
Spherical Mean

AN ANALOGUE OF COWLING–PRICE'S THEOREM AND HARDY'S THEOREM FOR THE GENERALIZED FOURIER TRANSFORM ASSOCIATED WITH THE SPHERICAL MEAN OPERATOR

2004 ◽  
Vol 02 (03) ◽  
pp. 177-192 ◽  
Author(s):  
C. CHETTAOUI ◽  
Y. OTHMANI ◽  
K. TRIMÈCHE
Keyword(s):  
Fourier Transform ◽  
Harmonic Analysis ◽  
Generalized Fourier Transform ◽  
Spherical Mean Operator ◽  
Hardy’S Theorem ◽  
Spherical Mean

Using the harmonic analysis associated with the spherical mean operator R and its dualtR we establish an analogue of Cowling–Price's theorem and Hardy's theorem for the spherical mean operator R.

Weighted inequalities and uncertainty principles for the (k,a)-generalized Fourier transform

2016 ◽  
Vol 27 (03) ◽  
pp. 1650019 ◽  
Author(s):  
Troels Roussau Johansen
Keyword(s):  
Fourier Transform ◽  
Young Inequality ◽  
Uncertainty Principles ◽  
Weighted Inequalities ◽  
Generalized Fourier Transform ◽  
Pitt’S Inequality ◽  
Uncertainty Inequalities

We obtain several versions of the Hausdorff–Young and Hardy–Littlewood inequalities for the [Formula: see text]-generalized Fourier transform recently investigated at length by Ben Saïd, Kobayashi, and Ørsted. We also obtain a number of weighted inequalities — in particular Pitt’s inequality — that have application to uncertainty principles. Specifically we obtain several analogs of the Heisenberg–Pauli–Weyl principle for [Formula: see text]-functions, local Cowling–Price-type inequalities, Donoho–Stark-type inequalities and qualitative extensions. We finally use the Hausdorff–Young inequality as a means to obtain entropic uncertainty inequalities.

Localization operators, time frequency concentration and quantitative-type uncertainty for the continuous wavelet transform associated with spherical mean operator

2019 ◽  
Vol 17 (04) ◽  
pp. 1950022 ◽  
Author(s):  
Hatem Mejjaoli ◽  
Nadia Ben Hamadi ◽  
Slim Omri
Keyword(s):  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Finite Measure ◽  
Continuous Wavelet ◽  
Uncertainty Principles ◽  
Localization Operators ◽  
Spherical Mean Operator ◽  
Von Neumann ◽  
Time Frequency ◽  
Spherical Mean

We consider the continuous wavelet transform [Formula: see text] associated with the spherical mean operator. We investigate the localization operators for [Formula: see text], in particular, we prove that they are in the Schatten-von Neumann class. Next, we analyze the concentration of this transform on sets of finite measure. In particular, Donoho-Stark and Benedicks-type uncertainty principles are given. Finally, we prove many versions of quantitative uncertainty principles for [Formula: see text].

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