Time-frequency analysis associated with the generalized Stockwell transform

Abstract

The Riemann-Liouville operator has been extensively investigated and has witnessed a remarkable development in numerous fields of harmonic analysis. In this paper, we consider the Stockwell transform associated with the Riemann-Liouville operator. Knowing the fact that the study of the time-frequency analysis are both theoretically interesting and practically useful, we investigated several problems for this subject on the setting of this generalized Stockwell transform. Firstly, we study the boundedness and compactness of localization operators associated with the generalized Stockwell transform. Next, we explore the Shapiro uncertainty principle for the previous transform. Finally, the scalogram for the generalized Stockwell transform are introduced and studied at the end.

Keywords

• Riemann-Liouville operator
• localization operator
• generalized Stockwell transform
• Shapiro’s theorem
• scalogram

References

1. [1] L. D. Abreu, K. Grochenig and J.L. Romero, On accumulated spectrograms, Trans. Amer. Math. Soc. 368, 3629–3649, 2016.
2. [2] P. S. Addison, J.N. Watson, G.R. Clegg, P.A. Steen, C.E. Robertson, Finding coordinated atrial activity during ventricular fibrilation using wavelt decomposition, analyzing surface ECGs with a new analysis technique to better understand cardiac death, IEEE Trans. Eng. Med. Biol. 21, 58–65, 2002.
3. [3] B. Amri, L.T. Rachdi, Beckner logarithmic uncertainty principle for the Riemann- Liouville operator, Internat. J. Math. 1350070, 24 (9), 1-29, 2013.
4. [4] C. Baccar, N. Ben Hamadi, L.T. Rachdi, Inversion formulas for the Riemann- Liouville transform and its dual associated with singular partial differential operators, Int. J. Math. Math. Sci. 2006, 86238, 1-26, 2006.
5. [5] C. Baccar, L. T. Rachdi, Spaces of DLp-type and a convolution product associated with the Riemann-Liouville operator, Bull. Math. Anal. Appl. 1 (3), 16–41, 2009.
6. [6] C. Baccar, N. Ben Hamadi, Localization operators of the wavelet transform associated to the Riemann-Liouville operator, Int. J. Math. 27, 1650036, 2016.
7. [7] C. Baccar, N. Ben Hamadi, H. Herch and F. Meherzi, Inversion of the Riemann- Liouville operator and its dual using wavelets, Opusc. Math. 35 (6), 2015.
8. [8] N. Ben Hamadi, Localization operators for the windowed Fourier transform associated with singular partial differential operators, Rocky Mt J. Math. 47 (7), 2179-2195, 2017.

9. [9] N. Ben Hamadi, A. Ghandouri and Z. Hafirassou Beckner Logarithmic Uncertainty Principle for the Stockwell Transform Associated with the Singular Partial Differential Operators, Mediterr. J. Math. 20 (4), 211, 2023.
10. [10] N. Ben Hamadi, Z. Hafirassou, and H. Herch, Uncertainty principles for the Hankel- Stockwell transform, J. Pseudo-Differ. Oper. Appl. 11, 543-564, 2020.
11. [11] H-P Cai , ZH. He, DJ. Huang, Seismic data denoising based on mixed time-Frequency methods, Appl Geophys. 8, 319-327, 2011.
12. [12] V. Catana, Schatten-von Neumann norm inequalities for two-wavelet localization operators associated with β-Stockwell transforms, Appl Anal. 91, 503-515, 2012.
13. [13] I. Djurovic, E., Sejdic and J. Jiang, Frequency-based window width optimization for S−transform, AEU Int J Electron Commun. 62, 245-250, 2008.
14. [14] J. Du, MW. Wong, and H. Zhu, Continuous and discrete inversion formulas for the Stockwell transform, Integr. Transforms Spec. Funct. 18, 537-543, 2007.
15. [15] J. A. Fawcett, Inversion of N-dimensional spherical means, SIAM. J.Appl. Math. 45, 336-341, 1983.
16. [16] 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), 429-441, 1946.
17. [17] H. Helesten, L. E. Andersson, An inverse method for the processing of synthetic aperture radar data, Inv. Prob. 3, 111-124, 1987.
18. [18] K. Hleili, S. Omri, L. Rachdi, Uncertainty principle for the Riemann-Liouville operator, Cubo (A Mathematical Journal) 13 (3), 91-115, 2015.
19. [19] F. John, Plane waves and spherical means applied to partial differential equations, Interscience, New York, 1955.
20. [20] L. Liu, A trace class operator inequality, J. Math. Anal. Appl. 328, 1484-1486, 2007.
21. [21] E. Malinnikova, Orthonormal sequences in L2(Rd) and time frequency localization, J. Fourier Anal. Appl. 16 (6), 983-1006, 2010.
22. [22] H. Mejjaoli, Spectral theorems associated with the Riemann-Liouville two-wavelet localization operators, Anal. Math. 45, 347-374, 2019.
23. [23] H. Mejjaoli, Y. Othmani, Qualitative and quantitative uncertainty principles associated with the Reimann-Liouville operator, LE MATEMATICHE Vol. LXXI, Fasc. II, 173-202, 2016.
24. [24] H. Mejjaoli, S. Omri, Boundedness and compactness of Reimann-Liouville two-wavelet multipliers, J. Pseudo-Differ. Oper. Appl. 9 (1), 189-213, 2018.
25. [25] H. Mejjaoli, K. Trimèche, Spectral theorems associated with the Riemann-Liouville- Wigner localization operators, Rocky Mountain J. Math. 49 (1), 247-281, 2019.
26. [26] H. Mejjaoli, Dunkl-Stockwell transform and its applications to the time-frequency analysis, J. Pseudo-Differ.Oper. Appl. 12 (2), 1-59, 2021.
27. [27] H. Mejjaoli, Quantitative uncertainty principles for the generalized Stockwell transform, To appear in Int. J. Open Problems Compt. Math.
28. [28] H. Mejjaoli, Time-frequency analysis associated with the generalized Stockwell transform in the spherical mean operator theory, To appear in Int. J. Open Problems Complex Analysis.
29. [29] L. Riba and MW. Wong, Continuous inversion formulas for multi-dimensional modified Stockwell transforms, Integral Transforms Spec. Func. 26 (1), 9-19, 2015.
30. [30] F. Riesz and B. Sz. Nagy, Functional Analysis, Frederick Ungar Publishing Co., New York, 1995.
31. [31] FA. Shah and AY Tantary, Non-isotropic angular Stockwell transform and the associated uncertainty principles, Appl Anal. 100 (4), 835-859, 2021.
32. [32] HM. Srivastava, FA. Shah and AY Tantary, A family of convolution-based generalized Stockwell transforms J. Pseudo-Differ. Oper. Appl. 11, 1505-1536, 2020.
33. [33] RG. Stockwell, L. Mansinha and RP. Lowe, Localization of the complex spectrum: The S Transform, IEEE Trans Signal Process 44, 998-1001, 1996.
34. [34] EM. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83, 482-49, 1956.
35. [35] P. Sukiennik, J. T. Bialasiewicz, Cross-correlation of bio-signals using continuous wavelet transform and genetic algorithm, J. Neurosci. Meth 247, 13-22, 2015.
36. [36] K. Trimèche, Permutation operators and the central limit theorem associated with partial differential operators, Proceedings of the tenth Oberwolfach conference on probability measures on groups, held November 4-10,1990 in Oberwolfach, Germany. Probability measures on groups X, 395-424, 1991.
37. [37] K. Trimèche, Generalized Wavelets and Hypergroups, Gordon and Breach Science Publishers, 1997.
38. [38] M. W. Wong, Wavelet transforms and localization operators, 136 Springer Science & Business Media, 2002.