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)
Finally, we prove an analogous inequality for the windowed Fourier transform.
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References
Andrews, G., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge (1999)
Donoho, D.L., Stark, P.B.: Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49(3), 906–931 (1989)
Fawcett, J.A.: Inversion of n-dimensional spherical means. SIAM J. Appl. Math. 45, 336–341 (1985)
Gabor, D.: Theory of communication. J. Inst. Electr. Eng. 93(26), 429–457 (1946)
Ghobber, S., Jaming, P.: Uncertainty principles for integral operators. Studia Math. 220, 197–220 (2014)
Gröchenig, K.H.: Foundations of time-frequency analysis. Applied and Numerical Harmonic Analysis. Birkhäuser Boston, Inc., Boston, MA (2001)
Hellsten, H., Andersson, L.E.: An inverse method for the processing of synthetic aperture radar data. Inverse Problems 3(1), 111–124 (1987)
Herberthson, M.: A numerical implementation of an inverse formula for CARABAS raw data. Internal report D 30430-3.2. national defense research institute, FOA Box 1165; S-581 11 Linköping, Sweden (1986)
Jaming, P., Powell, A.M.: Uncertainty principles for orthonormal sequences. J. Funct. Anal. 243, 611–630 (2007)
Jelassi, M., Rachdi, L.T.: On the range of the Fourier transform associated with the spherical mean operator. Fract. Calc. Appl. Anal. 7(4), 379–402 (2004)
Lebedev, N.N.: Special Functions and Their Applications. Dover Publications, New York (1972)
Malinnikova, E.: Orthonormal sequences in L2(Rd) and time frequency localization. J. Fourier Anal. Appl. 16, 983–1006 (2010)
Msehli, N., Rachdi, L.T.: Heisenberg-Pauli-Weyl uncertainty principle for the spherical mean operator. Mediterr. J. Math. 7, 169–194 (2010)
Nessibi, M.M., Rachdi, L.T., Trimèche, K.: Ranges and inversion formulas for spherical mean operator and its dual. J. Math. Anal. Appl. 196(3), 861–884 (1995)
Rachdi, L.T., Trimèche, K.: Weyl transforms associated with the spherical mean operator. Anal. Appl. (Singap.) 1(2), 141–164 (2003)
Saitoh, S.: Theory of reproducing kernels and its applications. Pitman Research Notes in Mathematics Series, 189. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York (1988)
Schuster, T.: The method of approximate inverse: theory and applications. Lecture Notes in Math., vol. 1906. Springer, Berlin (2007)
Shapiro, H.S.: Uncertainty principles for bases in \(l^{2} \mathbb {R}\). In: Proceedings of the Conference on Harmonic Analysis and Number Theory. CIRM. Marseille-Luminy, pp 16–21 (2005)
Trimèche, K.: Transformation intégrale de Weyl et théorème de Paley-Wiener associés à un opérateur différentiel singulier sur \((0,\infty )\). J. Math. Pures Appl. 60, 51–98 (1981)
Trimèche, K.: Inversion of the Lions translation operator using genaralized wavelets. Appl. Comput. Harmonic Anal. 4, 97–112 (1977)
Wang, L.V.: Photoacoustic imaging and spectroscopy. Optical science and engineering (Book 144) (2009)
Zhao, J.: Localization operators associated with the sphercial mean operator. Anal. Theory Appl. 21(4), 317–325 (2005)
Zhao, J., Peng, L.: Wavelet and Weyl transforms associated with the spherical mean operator. Integral Equations Operator Theory 50, 279–290 (2004)
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Hleili, K. Some Results for the Windowed Fourier Transform Related to the Spherical Mean Operator. Acta Math Vietnam 46, 179–201 (2021). https://doi.org/10.1007/s40306-020-00382-2
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DOI: https://doi.org/10.1007/s40306-020-00382-2
Keywords
- Spherical mean operator
- Windowed fourier transform
- Localization operators
- Shapiro’s uncertainty principles