Wavelet multipliers for the linear canonical deformed Hankel transform and applications

Saifallah Ghobber; Hatem Mejjaoli; Saifallah Ghobber; Hatem Mejjaoli

doi:10.3934/math.20251185

AIMS Mathematics

2025, Volume 10, Issue 11: 26958-26993. doi: 10.3934/math.20251185

Previous Article Next Article

Research article Special Issues

Wavelet multipliers for the linear canonical deformed Hankel transform and applications

Saifallah Ghobber ^{1
,
,},
Hatem Mejjaoli ²

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
2.
Department of Mathematics, College of Sciences, Taibah University, P.O. Box 30002, Al Madinah Al Munawarah 42353, Saudi Arabia

Received: 04 July 2025 Revised: 01 November 2025 Accepted: 11 November 2025 Published: 20 November 2025
MSC : 42B15, 44A05

In this paper, we introduce the notion of a wavelet multiplier in the setting of the linear canonical deformed Hankel transform (LCDHT), which depends on a symbol and two bounded functions. Then, we study the boundedness and compactness of these operators according to the symbol and the bounded functions. We will then show that, under certain assumptions, the wavelet multiplier is equal to the well-known time-frequency restriction operator. Then, we show that a function that is almost time- and band-limited can be approximated by its projection on the subspace spanned by the first eigenfunctions of such an operator, corresponding to the greatest eigenvalues, which are near one. This study for the LCDHT includes, in particular, some known transforms, such as the deformed Hankel, the Fresnel, and the fractional deformed Hankel transforms.
- linear canonical deformed Hankel transform,
- wavelet multipliers
Citation: Saifallah Ghobber, Hatem Mejjaoli. Wavelet multipliers for the linear canonical deformed Hankel transform and applications[J]. AIMS Mathematics, 2025, 10(11): 26958-26993. doi: 10.3934/math.20251185

Related Papers:

Abstract

In this paper, we introduce the notion of a wavelet multiplier in the setting of the linear canonical deformed Hankel transform (LCDHT), which depends on a symbol and two bounded functions. Then, we study the boundedness and compactness of these operators according to the symbol and the bounded functions. We will then show that, under certain assumptions, the wavelet multiplier is equal to the well-known time-frequency restriction operator. Then, we show that a function that is almost time- and band-limited can be approximated by its projection on the subspace spanned by the first eigenfunctions of such an operator, corresponding to the greatest eigenvalues, which are near one. This study for the LCDHT includes, in particular, some known transforms, such as the deformed Hankel, the Fresnel, and the fractional deformed Hankel transforms.

References

[1]	H. J. Landau, On Szegö's eigenvalue distribution theorem and non-Hermitian kernels, J. Anal. Math., 28 (1975), 335–357. https://doi.org/10.1007/BF02786820 doi: 10.1007/BF02786820
[2]	H. J. Landau, H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty–Ⅲ: The dimension of the space of essentially time- and band-limited signals, Bell Syst. Tech. J., 41 (1962), 1295–1336. https://doi.org/10.1002/j.1538-7305.1962.tb03279.x doi: 10.1002/j.1538-7305.1962.tb03279.x
[3]	D. L. Donoho, P. B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math., 49 (1989), 906–931. https://doi.org/10.1137/0149053 doi: 10.1137/0149053
[4]	Z. He, M. W. Wong, Wavelet multipliers and signals, J. Aust. Math. Soc. Ser. B, 40 (1999), 437–446. https://doi.org/10.1017/S0334270000010523 doi: 10.1017/S0334270000010523
[5]	J. Du, M. W. Wong, Traces of wavelet multipliers, Math. Rep. Acad. Sci, 23 (2001), 148–152.
[6]	M. W. Wong, Wavelet transforms and localization operators, Birkhäuser Basel, 2002. https://doi.org/10.1007/978-3-0348-8217-0
[7]	H. Mejjaoli, Boundedness and compactness of Dunkl two-wavelet multipliers, Int. J. Wavelets Multi., 15 (2017), 1750048. https://doi.org/10.1142/S0219691317500485 doi: 10.1142/S0219691317500485
[8]	S. Ghobber, Fourier-like multipliers and applications for integral operators, Complex Anal. Oper. Theory, 13 (2019), 1059–1092. https://doi.org/10.1007/s11785-018-0839-9 doi: 10.1007/s11785-018-0839-9
[9]	V. Catană, M. G. Scumpu, Localization operators and wavelet multipliers involving two-dimensional linear canonical curvelet transform, J. Pseudo-Differ. Oper. Appl., 14 (2023), 53. https://doi.org/10.1007/s11868-023-00547-1 doi: 10.1007/s11868-023-00547-1
[10]	S. Ghobber, A variant of the Hankel multiplier, Banach J. Math. Anal., 12 (2018), 144–166. https://doi.org/10.1215/17358787-2017-0051 doi: 10.1215/17358787-2017-0051
[11]	H. Mejjaoli, Spectral theorems associated with the $(k, a)$-generalized wavelet multipliers, J. Pseudo-Differ. Oper. Appl., 9 (2018), 735–762. https://doi.org/10.1007/s11868-018-0260-1 doi: 10.1007/s11868-018-0260-1
[12]	H. Mejjaoli, New results for the Hankel two-wavelet multipliers, J. Taibah Univ. Sci., 13(1) 1750048 (2019), 32-40. https://doi.org/10.1080/16583655.2018.1521711
[13]	H. Mejjaoli, K. Trimèche, Two-wavelet multipliers on the dual of the Laguerre hypergroup and applications, Mediterr. J. Math., 16 (2019), 126. https://doi.org/10.1007/s00009-019-1389-8 doi: 10.1007/s00009-019-1389-8
[14]	H. Mejjaoli, Wavelet-multipliers analysis in the framework of the $k$-Laguerre theory, Linear Multilinear A., 67 (2019), 70–93. https://doi.org/10.1080/03081087.2017.1410093 doi: 10.1080/03081087.2017.1410093
[15]	H. Mejjaoli, S. Omri, Spectral theorems associated with the directional short-time Fourier transform, J. Pseudo-Differ. Oper. Appl., 11 (2020), 15–54. https://doi.org/10.1007/s11868-019-00308-z doi: 10.1007/s11868-019-00308-z
[16]	P. Shkula, S. K. Upadhyay, Wavelet multiplier associated with the Watson transform, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 117 (2023), 15. https://doi.org/10.1007/s13398-022-01342-1 doi: 10.1007/s13398-022-01342-1
[17]	H. M. Srivastava, P. Shukla, S. K. Upadhyay, The localization operator and wavelet multipliers involving the Watson transform, J. Pseudo-Differ. Oper. Appl., 13 (2022), 46. https://doi.org/10.1007/s11868-022-00477-4 doi: 10.1007/s11868-022-00477-4
[18]	S. A. Collins, Lens-system diffraction integral written in terms of matrix optics, J. Opt. Soc. Am., 60 (1970), 1168–1177. https://doi.org/10.1364/JOSA.60.001168 doi: 10.1364/JOSA.60.001168
[19]	M. Moshinsky, C. Quesne, Linear canonical transformations and their unitary representations, J. Math. Phys., 12 (1971), 1772–1780. https://doi.org/10.1063/1.1665805 doi: 10.1063/1.1665805
[20]	A. Bultheel, H. Martínez-Sulbaran, Recent development in the theory of the fractional Fourier and linear canonical transforms, Bull. Belg. Math. Soc. Simon Stevin, 13 (2007), 971–1005. https://doi.org/10.36045/bbms/1170347822 doi: 10.36045/bbms/1170347822
[21]	B. Barshan, M. A. Kutay, H. M. Ozaktas, Optimal filtering with linear canonical transformations, Opt. Commun., 135 (1997), 32–36. https://doi.org/10.1016/S0030-4018(96)00598-6 doi: 10.1016/S0030-4018(96)00598-6
[22]	B. M. Hennelly, J. T. Sheridan, Fast numerical algorithm for the linear canonical transform, J. Opt. Soc. Amer. A, 22 (2005), 928–937. https://doi.org/10.1364/JOSAA.22.000928 doi: 10.1364/JOSAA.22.000928
[23]	J. J. Healy, M. A. Kutay, H. M. Ozaktas, J. T. Sheridan, Linear canonical transforms, New York: Springer, 2016. https://doi.org/10.1007/978-1-4939-3028-9
[24]	H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The fractional Fourier transform with applications in optics and signal processing, New York: John Wiley & Sons, 2001.
[25]	K. B. Wolf, Canonical transforms. Ⅱ. Complex radial transforms, J. Math. Phys., 15 (1974), 2102–2111. https://doi.org/10.1063/1.1666590 doi: 10.1063/1.1666590
[26]	T. Z. Xu, B. Z. Li, Linear canonical transform and its applications, Beijing: Science Press, 2013.
[27]	S. Ghazouani, F. Bouzeffour, Heisenberg uncertainty principle for a fractional power of the deformed Hankel transform on the real line, J. Comput. Appl. Math., 294 (2016), 151–176. https://doi.org/10.1016/j.cam.2015.06.013 doi: 10.1016/j.cam.2015.06.013
[28]	H. Mejjaoli, S. Negzaoui, Linear canonical deformed Hankel transform and the associated uncertainty principles, J. Pseudo-Differ. Oper. Appl., 14 (2023), 29. https://doi.org/10.1007/s11868-023-00518-6 doi: 10.1007/s11868-023-00518-6
[29]	F. A. Shah, A. Y. Tantary, Multi-dimensional linear canonical transform with applications to sampling and multiplicative filtering, Multidim. Syst. Sign. Process., 33 (2022), 621–650. https://doi.org/10.1007/s11045-021-00816-6 doi: 10.1007/s11045-021-00816-6
[30]	F. A. Shah, A. Y. Tantary, Linear canonical ripplet transform: Theory and localization operators, J. Pseudo-Differ. Oper. Appl., 13 (2022), 45. https://doi.org/10.1007/s11868-022-00476-5 doi: 10.1007/s11868-022-00476-5
[31]	S. Ghobber, H. Mejjaoli, Localization operators for the linear canonical Dunkl windowed transformation, Axioms, 14 (2025), 262. https://doi.org/10.3390/axioms14040262 doi: 10.3390/axioms14040262
[32]	S. Ghobber, H. Mejjaoli, Novel Gabor-type transform and weighted uncertainty principles, Mathematics, 13 (2025), 1109. https://doi.org/10.3390/math13071109 doi: 10.3390/math13071109
[33]	H. Yang, Q. Feng, X. Wang, D. Urynbassarova, A. A. Teali, Reduced biquaternion windowed linear canonical transform: Properties and applications, Mathematics, 12 (2024), 743. https://doi.org/10.3390/math12050743 doi: 10.3390/math12050743
[34]	M. Bahri, S. A. A. Karim, Novel uncertainty principles concerning linear canonical wavelet transform, Mathematics, 10 (2022), 3502. https://doi.org/10.3390/math10193502 doi: 10.3390/math10193502
[35]	S. Ghobber, H. Mejjaoli, A new wavelet transform and its localization operators, Mathematics, 13 (2025), 1771. https://doi.org/10.3390/math13111771 doi: 10.3390/math13111771
[36]	J. Shi, X. Liu, N. Zhang, Generalized convolution and product theorems associated with linear canonical transform, SIViP, 8 (2014), 967–974. https://doi.org/10.1007/s11760-012-0348-7 doi: 10.1007/s11760-012-0348-7
[37]	D. Urynbassarova, A. A. Teali, Convolution, correlation, and uncertainty principles for the quaternion offset linear canonical transform, Mathematics, 11 (2023), 2201. https://doi.org/10.3390/math11092201 doi: 10.3390/math11092201
[38]	M. Bahri, S. A. A Karim, B. A. S., M. Nur, N. Nurwahidah, A new form of convolution theorem for one-dimensional quaternion linear canonical transform and application, Symmetry, 17 (2025), 1004. https://doi.org/10.3390/sym17071004 doi: 10.3390/sym17071004
[39]	S. Ghazouani, E. A. Soltani, A. Fitouhi, A unified class of integral transforms related to the deformed Hankel transform, J. Math. Anal. Appl., 449 (2017), 1797–1849. https://doi.org/10.1016/j.jmaa.2016.12.054 doi: 10.1016/j.jmaa.2016.12.054
[40]	J. F. Zhang, S. P. Hou, The generalization of the Poisson sum formula associated with the linear canonical transform, J. Appl. Math., 2012 (2012), 102039. https://doi.org/10.1155/2012/102039 doi: 10.1155/2012/102039
[41]	A. A. Teali, F. A. Shah, Wave packet frames in linear canonical domains: Construction and perturbation, J. Pseudo-Differ. Oper. Appl., 15 (2024), 74. https://doi.org/10.1007/s11868-024-00645-8 doi: 10.1007/s11868-024-00645-8
[42]	S. B. Saïd, T. Kobayashi, B. Ørsted, Laguerre semigroup and Dunkl operators, Compos. Math., 148 (2012), 1265–1336. https://doi.org/10.1112/S0010437X11007445 doi: 10.1112/S0010437X11007445
[43]	V. Havin, B. Jöricke, The uncertainty principle in harmonic analysis, Berlin: Springer-Verlag, 1994. https://doi.org/10.1007/978-3-642-78377-7
[44]	L. D. Abreu, J. M. Pereira, Measures of localization and quantitative Nyquist densities, Appl. Comput. Harmon. Anal., 38 (2015), 524–534. https://doi.org/10.1016/j.acha.2014.08.002 doi: 10.1016/j.acha.2014.08.002
[45]	J. F. Diejen, L. Vinet, Calogero-Moser-Sutherland models, New York: Springer, 2000. https://doi.org/10.1007/978-1-4612-1206-5
[46]	C. Bennett, R. Sharpley, Interpolation of operators, Academic Press, 1988.
[47]	J. Calderon, Intermediate spaces and interpolation, the complex method, Studia Math., 24 (1964), 113–190. https://doi.org/10.4064/sm-24-2-113-190 doi: 10.4064/sm-24-2-113-190
[48]	H. Wang, Compressed sensing: Theory and applications, J. Phys. Conf. Ser., 2419 (2023), 012042. https://doi.org/10.1088/1742-6596/2419/1/012042 doi: 10.1088/1742-6596/2419/1/012042
[49]	Y. An, Z. Xue, J. Ou, Deep learning-based sparsity-free compressive sensing method for high accuracy structural vibration response reconstruction, Mech. Syst. Signal Pr., 211 (2024), 111168. https://doi.org/10.1016/j.ymssp.2024.111168 doi: 10.1016/j.ymssp.2024.111168
[50]	M. Doi, M. Ohzeki, Phase transition in binary compressed sensing based on $L^{1}$-norm minimization, J. Phys. Soc. Jpn., 93 (2024), 084003. https://doi.org/10.7566/JPSJ.93.084003 doi: 10.7566/JPSJ.93.084003
[51]	L. R. Chandran, I. Karuppasamy, M. G. Nair, H. Sun, P. K. Krishnakumari, Compressive sensing in power engineering: A comprehensive survey of theory and applications, and a case study, J. Sens. Actuator Netw., 14 (2025), 28. https://doi.org/10.3390/jsan14020028 doi: 10.3390/jsan14020028
[52]	L. Zhao, Y. Zhang, X. Wang, J. Zhang, H. Bai, A. Wang, A survey on image compressive sensing: From classical theory to the latest explicable deep learning, Pattern Recogn., 170 (2026), 112022. https://doi.org/10.1016/j.patcog.2025.112022 doi: 10.1016/j.patcog.2025.112022
[53]	S. Ghobber, P. Jaming, Strong annihilating pairs for the Fourier-Bessel transform, J. Math. Anal. Appl., 377 (2011), 501–515. https://doi.org/10.1016/j.jmaa.2010.11.015 doi: 10.1016/j.jmaa.2010.11.015
[54]	G. A. M. Velasco, M. Dörfler, Sampling time-frequency localized functions and constructing localized time-frequency frames, Eur. J. Appl. Math., 28 (2017), 854–876. https://doi.org/10.1017/S095679251600053X doi: 10.1017/S095679251600053X

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)