Uncertainty principles for the \(q\)-Bessel windowed transform and localization operators. (English) Zbl 1533.42008
Author’s abstract: In this work, considering the \(q\)-harmonic analysis associated with the \(q\)-Bessel Fourier operator of order zero, for a fixed \(q\in ]0,1[\), we define the \(q\)-Bessel windowed transform and show a \(q\)-analogue of the usual inversion formula. Then, Schatten-class properties of localization operators associated with the \(q\)-Bessel windowed transform are studied. A new version of Heisenberg’s uncertainty principles and concentration-type principles for the \(q\) windowed Bessel Fourier transform is also proved. Finally, we will show that these uncertainty principles can be refined for orthonormal sequences.
Reviewer: Sorin Gheorghe Gal (Oradea)
MSC:
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 05A30 | \(q\)-calculus and related topics |
| 33D15 | Basic hypergeometric functions in one variable, \({}_r\phi_s\) |
Keywords:
\(q\)-Bessel Fourier transform; \(q\)-Bessel windowed transform; uncertainty principles; time-frequency concentration; localization operators; Schatten-von Neumann classReferences:
| [1] | N. Bettaibi, F. Bouzeffour, H. Ben Elmonser and W. Binous, Elements of harmonic analysis related to the third basic zero order Bessel function.. J. Math. Anal. Appl., 324(2008) 1203-1219. · Zbl 1154.33007 |
| [2] | Cohen, L., Generalized phase-space distribution functions, J. Math. Phys, 7, 781-786 (1966) · doi:10.1063/1.1931206 |
| [3] | Cohen, L., Time-frequency distributions-a review. Pro. IEEE, 77, 941-981 (1989) |
| [4] | Debnath, L.; Shah, FA, Lectuer Notes on Wavelet Transforms (2017), Boston: Birkhäuser, Boston · Zbl 1379.42001 · doi:10.1007/978-3-319-59433-0 |
| [5] | Dhaouadi, L., On the \(q\)-Bessel Fourier transform, Bull. Math. Anal. Appl., 5, 2, 42-60 (2013) · Zbl 1314.33016 |
| [6] | Dhaouadi, L.; Binous, W.; Fitouhi, A., Paley-Wiener theorem for the \(q\)-Bessel Fourier transform and associated q-sampling formula, Expos. Math, 27, 1, 55-72 (2009) · Zbl 1172.33308 · doi:10.1016/j.exmath.2008.07.002 |
| [7] | Fitouhi, A.; Dhaoudi, L., Positivity of the generalized translation associated with the \(q\)-Hankel transform, Constr. Approx., 34, 435-472 (2011) · Zbl 1247.33030 · doi:10.1007/s00365-011-9132-0 |
| [8] | Fitouhi, A.; Hamza, MM; Bouzeffour, F., The \(q-J_{\alpha }\) Bessel function, J. Approx. Theory., 115, 144-166 (2002) · Zbl 1003.33007 · doi:10.1006/jath.2001.3645 |
| [9] | D. Gabor, Theory of communication. part 1: The analysis of information. Journal of the Institution of Electrical Engineers-Part III: Radio and Communication Engineering, ;93(26) (1946): 429-441. |
| [10] | Gasper, G.; Rahman, M., Basic Hypergeometric Series Encyclopedia of Mathematics and its Application, 35 (1990), Cambridge: Cambridge University Press, Cambridge · Zbl 0695.33001 |
| [11] | S. Ghobber, Time-frequency concentration and localization operators in the Dunkl setting, 7(3) (2016): 431-449. J. Pseudo-Differ. Oper. Appl. · Zbl 1358.42031 |
| [12] | Jackson, FH, On \(q-\) definite integrals Quart, J. Pur. Appl. Maths., 41, 193-203 (1910) · JFM 41.0317.04 |
| [13] | T. H. Koorwinder and R. F. Swarttouw, On \(q-\) analogues of the Fourier and Hankel transforms, Trans. Amer. Soc, 333, (1992): 445-461. · Zbl 0759.33007 |
| [14] | Liu, L., A trace class operator inequality, J. Math. Anal. Appl., 328, 1484-1486 (2007) · Zbl 1114.47020 · doi:10.1016/j.jmaa.2006.04.092 |
| [15] | E. Malinnikova, Orthonormal sequences in \(L^2(\mathbb{R}^d)\) and time frequency localization. J. Fourier. Anal. Appl 16 (2010): 983-1006. · Zbl 1210.42020 |
| [16] | H. Mejjaoli, \(k\)-Hankel Gabor transform on \(\mathbb{R}^d\) and its Applications to the Reproducing Kernel Theory. Complex Anal. Oper. Theory. 47(3) (2021): 1-35. |
| [17] | H. Mejjaoli, Practical inversion formulas for the Dunkl-Gabor transform on \(\mathbb{R}^d \). Integral Transforms. Spec. Funct. 33 (2012): 875-890. · Zbl 1257.35005 |
| [18] | Mejjaoli, H.; Sraieb, N., Gabor transform in quantum calculus and applications, Fractional Calculus and Applied Analysis., 12, 3, 319-336 (2009) · Zbl 1182.33026 |
| [19] | Mejjaoli, H.; Sraieb, N., Wavelet-multipliers analysis in the framework of the q-Dunkl theory, Int. J. Open Problems Complex Analysis., 11, 3, 1-27 (2019) |
| [20] | H. Mejjaoli and N. Sraieb, Time frequency analysis associated with the q-Bessel Stockwell transform. Preprint (2021) · Zbl 1482.44008 |
| [21] | H. Mejjaoli and N. Sraieb, The uncertainty principles for the q-Dunkl Gabor transform. Preprint (2022). · Zbl 1181.26036 |
| [22] | Mejjaoli, H.; Sraieb, N., The q-Dunkl wavelet theory and localization operators, Int. J. Open Problems Compt. Math., 13, 3, 106-127 (2020) |
| [23] | Mejjaoli, H.; Sraieb, N., Localization operators associated with the q-Bessel wavelet transform and applications, International Journal of Wavelets, Multiresolution and Information Processing., 19, 4, 1-33 (2021) · Zbl 1482.44008 · doi:10.1142/S0219691320500940 |
| [24] | A. P. Polychronakos, Exchange operator formalism for integrable systems of particles.Phys. Rev. Lett., 69 (1992). · Zbl 0968.37521 |
| [25] | J.F. Price, Inequalities and local uncertainty principles, Math. Phys.,: 24 (1978): 1711-1714. · Zbl 0513.60100 |
| [26] | N. Sraieb, \(k\)-Hankel Wigner transform and its applications to the Localization operators theory. J. Pseudo-Differ. Oper. Appl. 13 Juillet 2022., 13(3) (2022): 1-34. · Zbl 1518.43001 |
| [27] | Stein, EM, Interpolation of linear operators, Trans. Amer. Math. Soc., 83, 482-492 (1956) · Zbl 0072.32402 · doi:10.1090/S0002-9947-1956-0082586-0 |
| [28] | K. Trimèche, Generalized Wavelets and Hypergroups. Gordon and Breach Science Publishers, (1997). · Zbl 0926.42016 |
| [29] | Tyr, O.; Daher, R., Two-wavelet theory and two-wavelet localization operators on the \(q\)-Dunkl harmonic analysis, Assian-European Journal of Mathematics., 15, 12, 1-24 (2022) · Zbl 1504.42029 |
| [30] | M.W. Wong,Wavelet transforms and localization operators, 136, Springer Science & Business Media, 2002. · Zbl 1016.42017 |
| [31] | Zayed, AI, Function and Generalized Function Transformations (1996), Boca Raton, FL: CRC Press, Boca Raton, FL · Zbl 0851.44002 |
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