Generalized Stockwell transforms: Spherical mean operators and applications
-
Saifallah Ghobber
and
Hatem Mejjaoli
Abstract
The spherical mean operator has been widely studied and has seen remarkable development in many areas of harmonic analysis. In this paper, we consider the Stockwell transform related to the spherical mean operator. Since the study of time-frequency analysis is both theoretically interesting and practically useful, we will study several problems for the generalized Stockwell transform. Firstly, we explore the Shapiro uncertainty principle for this transformation. Next, we will study the boundedness and then the compactness of localization operators related to the generalized Stockwell transform, and finally we will introduce and study its scalogram.
Acknowledgements
The second author dedicated this paper to Professor Mohamed Selmi for his support and help, and for his distinguished career in mathematics, and especially for his contribution to the Tunisian School in Potential theory. The authors are deeply indebted to the referees for providing constructive comments which improved the contents of this article.
References
[1] L. D. Abreu, K. Gröchenig and J. L. Romero, On accumulated spectrograms, Trans. Amer. Math. Soc. 368 (2016), no. 5, 3629–3649. 10.1090/tran/6517Search in Google Scholar
[2] P. S. Addison, J. N. Watson, G. R. Clegg, P. A. Steen and C. E. Robertson, Finding coordinated atrial activity during ventricular fibrillation using wavelet decomposition, IEEE Eng. Medicine Biol. Mag. 21 (2002), no. 1, 58–65. 10.1109/51.993194Search in Google Scholar PubMed
[3] C. Baccar and N. B. Hamadi, Localization operators of the wavelet transform associated to the Riemann–Liouville operator, Internat. J. Math. 27 (2016), no. 4, Article ID 1650036. 10.1142/S0129167X16500361Search in Google Scholar
[4] C. Baccar, N. B. Hamadi and L. T. Rachdi, Inversion formulas for Riemann–Liouville transform and its dual associated with singular partial differential operators, Int. J. Math. Math. Sci. 2006 (2006), Article ID 86238. 10.1155/IJMMS/2006/86238Search in Google Scholar
[5] V. Catană, Schatten–von Neumann norm inequalities for two-wavelet localization operators associated with β-Stockwell transforms, Appl. Anal. 91 (2012), no. 3, 503–515. 10.1080/00036811.2010.549478Search in Google Scholar
[6] E. Cordero and K. Gröchenig, Time-frequency analysis of localization operators, J. Funct. Anal. 205 (2003), no. 1, 107–131. 10.1016/S0022-1236(03)00166-6Search in Google Scholar
[7] I. Daubechies, Time-frequency localization operators: A geometric phase space approach, IEEE Trans. Inform. Theory 34 (1988), no. 4, 605–612. 10.1109/18.9761Search in Google Scholar
[8] L. Debnath and F. A. Shah, Lecture Notes on Wavelet Transforms, Compact Textb. Math., Birkhäuser/Springer, Cham, 2017. 10.1007/978-3-319-59433-0Search in Google Scholar
[9] J. A. Fawcett, Inversion of n-dimensional spherical averages, SIAM J. Appl. Math. 45 (1985), no. 2, 336–341. 10.1137/0145018Search in Google Scholar
[10] S. Ghobber, Some results on wavelet scalograms, Int. J. Wavelets Multiresolut. Inf. Process. 15 (2017), no. 3, Article ID 17500191. 10.1142/S0219691317500199Search in Google Scholar
[11] K. Gröchenig, Foundations of Time-Frequency Analysis, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, 2001. 10.1007/978-1-4612-0003-1Search in Google Scholar
[12] Z. He and M. W. Wong, Localization operators associated to square integrable group representations, PanAmer. Math. J. 6 (1996), no. 1, 93–104. Search in Google Scholar
[13] H. Hellsten and L.-E. Andersson, An inverse method for the processing of synthetic aperture radar data, Inverse Problems 3 (1987), no. 1, 111–124. 10.1088/0266-5611/3/1/013Search in Google Scholar
[14] D. Jhanwar, K. K. Sharma and S. G. Modani, Generalized fractional S-transform and its application to discriminate environmental background acoustic noise signals, Acoust. Phys. 60 (2014), 466–473. 10.1134/S1063771014040058Search in Google Scholar
[15] F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience, New York, 1955. Search in Google Scholar
[16] L. Liu, A trace class operator inequality, J. Math. Anal. Appl. 328 (2007), no. 2, 1484–1486. 10.1016/j.jmaa.2006.04.092Search in Google Scholar
[17]
E. Malinnikova,
Orthonormal sequences in
[18] L. Mansinha, R. Stockwell, R. Lowe, M. Eramian and R. Schincariol, Local S-spectrum analysis of 1-D and 2-D data, Phys. Earth Planet. Inter. 103 (1997), 329–336. 10.1016/S0031-9201(97)00047-2Search in Google Scholar
[19] H. Mejjaoli, Spectral theorems associated with the Riemann–Liouville two-wavelet localization operators, Anal. Math. 45 (2019), no. 2, 347–374. 10.1007/s10476-018-0609-ySearch in Google Scholar
[20] H. Mejjaoli, Dunkl–Stockwell transform and its applications to the time-frequency analysis, J. Pseudo-Differ. Oper. Appl. 12 (2021), no. 2, Paper No. 32. 10.1007/s11868-021-00378-ySearch in Google Scholar
[21] H. Mejjaoli and K. Trimeche, Spectral theorems associated with the Riemann–Liouville–Wigner localization operators, Rocky Mountain J. Math. 49 (2019), no. 1, 247–281. 10.1216/RMJ-2019-49-1-247Search in Google Scholar
[22] H. Mejjaoli and K. Trimèche, Quantitative uncertainty principles associated with the k-generalized Stockwell transform, Mediterr. J. Math. 19 (2022), no. 4, Paper No. 150. 10.1007/s00009-021-01968-2Search in Google Scholar
[23] M. M. Nessibi, L. T. Rachdi and K. Trimeche, Ranges and inversion formulas for spherical mean operator and its dual, J. Math. Anal. Appl. 196 (1995), no. 3, 861–884. 10.1006/jmaa.1995.1448Search in Google Scholar
[24] L. T. Rachdi and K. Trimèche, Weyl transforms associated with the spherical mean operator, Anal. Appl. (Singap.) 1 (2003), no. 2, 141–164. 10.1142/S0219530503000156Search in Google Scholar
[25] L. Riba and M. W. Wong, Continuous inversion formulas for multi-dimensional modified Stockwell transforms, Integral Transforms Spec. Funct. 26 (2015), no. 1, 9–19. 10.1080/10652469.2014.961452Search in Google Scholar
[26] F. Riesz and B. Sz. -Nagy, Functional Analysis, Frederick Ungar, New York, 1955. Search in Google Scholar
[27] F. A. Shah and A. Y. Tantary, Non-isotropic angular Stockwell transform and the associated uncertainty principles, Appl. Anal. 100 (2021), no. 4, 835–859. 10.1080/00036811.2019.1622681Search in Google Scholar
[28] S. Slim, Uncertainty principle in terms of entropy for the spherical mean operator, J. Math. Inequal. 5 (2011), no. 4, 473–489. 10.7153/jmi-05-42Search in Google Scholar
[29] H. M. Srivastava, F. A. Shah and A. Y. Tantary, A family of convolution-based generalized Stockwell transforms, J. Pseudo-Differ. Oper. Appl. 11 (2020), no. 4, 1505–1536. 10.1007/s11868-020-00363-xSearch in Google Scholar
[30] R. G. Stockwell, L. Mansinha and R. P. Lowe, Localization of the complex spectrum, the S transform, IEEE Trans. Signal Process. 44 (1996), no. 4, 998–1001. 10.1109/78.492555Search in Google Scholar
[31] P. Sukiennik and J. T. Bialasiewicz, Cross-correlation of bio-signals using continuous wavelet transform and genetic algorithm, J. Neurosci. Methods 247 (2015), 13–22. 10.1016/j.jneumeth.2015.03.002Search in Google Scholar PubMed
[32] K. Trimèche, Generalized Wavelets and Hypergroups, Gordon and Breach Science, Amsterdam, 1997. Search in Google Scholar
[33] M. W. Wong, Wavelet Transforms and Localization Operators, Oper. Theory Adv. Appl. 136, Birkhäuser, Basel, 2002. 10.1007/978-3-0348-8217-0Search in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Fractional p-Laplacian elliptic Dirichlet problems
- Group invertibility of the sum in rings and its applications
- σ-symmetric amenability of Banach algebras
- Generalized Stockwell transforms: Spherical mean operators and applications
- Existence of positive weak solutions for stationary fractional Laplacian problem by using sub-super solutions
- Capacity in Besov and Triebel–Lizorkin spaces with generalized smoothness
- Existence of solutions for (p(y),q(y))-Laplacian elliptic problem on an exterior domain
- Degeneration phenomenon in linear ordinary differential equations
- Weak type estimates of genuine Calderón–Zygmund operators on the local Morrey spaces associated with ball quasi-Banach function spaces
- Generalized derivation on semiprime and prime Banach algebras
- Weak positive solutions to singular quasilinear elliptic equation
- Constructions and characterizations of mixed reverse-order laws for the Moore–Penrose inverse and group inverse
- Integral inequalities of Ostrowski type for two kinds of s-logarithmically convex functions
- On the polar dualities and star dualities of the quasi Lp -intersection bodies
- New estimates for the Berezin number of Hilbert space operators
- Addendum to On Kuratowski partitions in the Marczewski and Laver structures and Ellentuck topology
Articles in the same Issue
- Frontmatter
- Fractional p-Laplacian elliptic Dirichlet problems
- Group invertibility of the sum in rings and its applications
- σ-symmetric amenability of Banach algebras
- Generalized Stockwell transforms: Spherical mean operators and applications
- Existence of positive weak solutions for stationary fractional Laplacian problem by using sub-super solutions
- Capacity in Besov and Triebel–Lizorkin spaces with generalized smoothness
- Existence of solutions for (p(y),q(y))-Laplacian elliptic problem on an exterior domain
- Degeneration phenomenon in linear ordinary differential equations
- Weak type estimates of genuine Calderón–Zygmund operators on the local Morrey spaces associated with ball quasi-Banach function spaces
- Generalized derivation on semiprime and prime Banach algebras
- Weak positive solutions to singular quasilinear elliptic equation
- Constructions and characterizations of mixed reverse-order laws for the Moore–Penrose inverse and group inverse
- Integral inequalities of Ostrowski type for two kinds of s-logarithmically convex functions
- On the polar dualities and star dualities of the quasi Lp -intersection bodies
- New estimates for the Berezin number of Hilbert space operators
- Addendum to On Kuratowski partitions in the Marczewski and Laver structures and Ellentuck topology
