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A new class of uncertainty principles for the k-Hankel wavelet transform

  • Hatem Mejjaoli ORCID logo and Firdous A. Shah ORCID logo EMAIL logo
Published/Copyright: March 31, 2023

Abstract

The k-Hankel wavelet transform is a novel addition to the class of wavelet transforms which relies on a pair of generalized translation and dilation operators governed by the well-known k-Hankel transform. The aim of this paper is to explore a class of new uncertainty principles associated with the k-Hankel wavelet transform, including the Benedick–Amrein–Berthier and Shapiro’s uncertainty inequalities. Nevertheless, we shall also establish certain local-type uncertainty principles abreast of the mean dispersion theorems for the k-Hankel wavelet transform.


Communicated by Siegfried Echterhoff


Acknowledgements

The authors are deeply indebted to the referees for constructive comments and help in improving the content of this article. The first author dedicates this paper to Professor Khalifa Trimèche and thanks Professor Man Wah Wong for his help.

References

[1] W. Beckner, Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1897–1905. 10.1090/S0002-9939-1995-1254832-9Search in Google Scholar

[2] S. Ben Saïd and L. Deleaval, Translation operator and maximal function for the (k,1)-generalized Fourier transform, J. Funct. Anal. 279 (2020), no. 8, Article ID 108706. 10.1016/j.jfa.2020.108706Search in Google Scholar

[3] S. Ben Saïd, T. Kobayashi and B. Ørsted, Laguerre semigroup and Dunkl operators, Compos. Math. 148 (2012), no. 4, 1265–1336. 10.1112/S0010437X11007445Search in Google Scholar

[4] M. Benedicks, On Fourier transforms of functions supported on sets of finite Lebesgue measure, J. Math. Anal. Appl. 106 (1985), no. 1, 180–183. 10.1016/0022-247X(85)90140-4Search in Google Scholar

[5] L. Debnath and F. A. Shah, Wavelet Transforms and Their Applications, 2nd ed., Birkhäuser/Springer, New York, 2015. 10.1007/978-0-8176-8418-1Search in Google Scholar

[6] 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

[7] C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), no. 1, 167–183. 10.1090/S0002-9947-1989-0951883-8Search in Google Scholar

[8] G. B. Folland and A. Sitaram, The uncertainty principle: A mathematical survey, J. Fourier Anal. Appl. 3 (1997), no. 3, 207–238. 10.1007/BF02649110Search in Google Scholar

[9] S. Ghobber and P. Jaming, Uncertainty principles for integral operators, Studia Math. 220 (2014), no. 3, 197–220. 10.4064/sm220-3-1Search in Google Scholar

[10] D. V. Gorbachev, V. I. Ivanov and S. Y. Tikhonov, Pitt’s inequalities and uncertainty principle for generalized Fourier transform, Int. Math. Res. Not. IMRN 2016 (2016), no. 23, 7179–7200. 10.1093/imrn/rnv398Search in Google Scholar

[11] V. P. Havin and B. Jöricke, The Uncertainty Principle in Harmonic Analysis, Springer, Berlin, 1994. 10.1007/978-3-642-78377-7Search in Google Scholar

[12] T. R. Johansen, Weighted inequalities and uncertainty principles for the ( k , a ) -generalized Fourier transform, Internat. J. Math. 27 (2016), no. 3, Article ID 1650019. 10.1142/S0129167X16500191Search in Google Scholar

[13] E. Malinnikova, Orthonormal sequences in L 2 ( 𝐑 d ) and time frequency localization, J. Fourier Anal. Appl. 16 (2010), no. 6, 983–1006. 10.1007/s00041-009-9114-9Search in Google Scholar

[14] H. Mejjaoli, Spectral theorems associated with the ( k , a ) -generalized wavelet multipliers, J. Pseudo-Differ. Oper. Appl. 9 (2018), no. 4, 735–762. 10.1007/s11868-018-0260-1Search in Google Scholar

[15] H. Mejjaoli, New results for the Hankel two-wavelet multipliers, J. Taibah Univ. Sci. 13 (2019), 32–40. 10.1080/16583655.2018.1521711Search in Google Scholar

[16] H. Mejjaoli, ( k , a ) -generalized wavelet transform and applications, J. Pseudo-Differ. Oper. Appl. 11 (2020), no. 1, 55–92. 10.1007/s11868-019-00291-5Search in Google Scholar

[17] H. Mejjaoli, Time-frequency analysis associated with the k-Hankel Gabor transform on d , J. Pseudo-Differ. Oper. Appl. 12 (2021), no. 3, Paper No. 41. 10.1007/s11868-021-00399-7Search in Google Scholar

[18] H. Mejjaoli, New uncertainty principles for the ( k , a ) -generalized wavelet transform, Rev. Un. Mat. Argentina 63 (2022), no. 1, 239–279. 10.33044/revuma.2051Search in Google Scholar

[19] H. Mejjaoli and F. A. Shah, Uncertainty principles associated with the directional short-time Fourier transform, J. Math. Phys. 62 (2021), no. 6, Paper No. 063511. 10.1063/5.0046426Search in Google Scholar

[20] H. Mejjaoli and N. Sraieb, Uncertainty principles for the continuous Dunkl Gabor transform and the Dunkl continuous wavelet transform, Mediterr. J. Math. 5 (2008), no. 4, 443–466. 10.1007/s00009-008-0161-2Search in Google Scholar

[21] H. Mejjaoli and K. Trimèche, k-Hankel two-wavelet theory and localization operators, Integral Transforms Spec. Funct. 31 (2020), no. 8, 620–644. 10.1080/10652469.2020.1723011Search in Google Scholar

[22] M. Rösler, Positivity of Dunkl’s intertwining operator, Duke Math. J. 98 (1999), no. 3, 445–463. 10.1215/S0012-7094-99-09813-7Search in Google Scholar

[23] S. Saitoh, Integral Transforms, Reproducing Kernels and Their Applications, Pitman Res. Notes Math. Ser. 369, Longman, Harlow, 1997. Search in Google Scholar

[24] F. A. Shah, K. S. Nisar, W. Z. Lone and A. Y. Tantary, Uncertainty principles for the quadratic-phase Fourier transform, Math. Methods Appl. Sci. 44 (2021), no. 13, 10416–10431. 10.1002/mma.7417Search in Google Scholar

[25] K. Trimèche, Generalized Wavelets and Hypergroups, Gordon and Breach Science, Amsterdam, 1997. Search in Google Scholar

[26] 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

Received: 2022-04-30
Revised: 2022-12-13
Published Online: 2023-03-31
Published in Print: 2023-05-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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