Abstract
In this paper, we prove a Heisenberg uncertainty inequality and a local uncertainty inequality for the Dunkl–Weinstein–Stockwell transform. We give also a Shapiro-type uncertainty inequality for this transform.
Acknowledgements
The authors are deeply grateful to the referees for their constructive comments and valuable suggestions.
References
[1] N. Ben Hamadi, Z. Hafirassou and H. Herch, Uncertainty principles for the Hankel–Stockwell transform, J. Pseudo-Differ. Oper. Appl. 11 (2020), no. 2, 543–564. 10.1007/s11868-020-00329-zSearch in Google Scholar
[2] N. Ben Hamadi, Z. Hafirassou and H. Mejjaoli, Time-frequency analysis associated with the generalized Stockwell transform, Hacet. J. Math. Stat. 53 (2024), no. 3, 748–776. 10.15672/hujms.1198408Search in Google Scholar
[3] N. Ben Salem, Hardy–Littlewood–Sobolev type inequalities associated with the Weinstein operator, Integral Transforms Spec. Funct. 31 (2020), no. 1, 18–35. 10.1080/10652469.2019.1652824Search in Google Scholar
[4] N. Ben Salem and A. R. Nasr, Shapiro type inequalities for the Weinstein and the Weinstein–Gabor transforms, Konuralp J. Math. 5 (2017), no. 1, 68–76. Search in Google Scholar
[5] P. C. Bowie, Uncertainty inequalities for Hankel transforms, SIAM J. Math. Anal. 2 (1971), 601–606. 10.1137/0502059Search in Google Scholar
[6] 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
[7] C. F. Dunkl, Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), no. 6, 1213–1227. 10.4153/CJM-1991-069-8Search in Google Scholar
[8] C. F. Dunkl, Hankel transforms associated to finite reflection groups, Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, Contemp. Math. 138, American Mathematical Society, Providence (1992), 123–138. 10.1090/conm/138/1199124Search in Google Scholar
[9] N. Fabio, The uncertainty principle for the short-time Fourier transform on finite cyclic groups: Cases of equality, J. Funct. Anal. 284 (2023), no. 12, Article ID 109924. 10.1016/j.jfa.2023.109924Search in Google Scholar
[10] W. G. Faris, Inequalities and uncertainty principles, J. Math. Phys. 19 (1978), no. 2, 461–466. 10.1063/1.523667Search in Google Scholar
[11] S. Ghobber and H. Mejjaoli, Reproducing kernel theory associated with the generalized Stockwell transform and applications, Complex Anal. Oper. Theory 17 (2023), no. 7, Paper No. 106. 10.1007/s11785-023-01407-ySearch in Google Scholar
[12] I. Gohberg, S. Goldberg and N. Krupnik, Traces and Determinants of Linear Operators, Oper. Theory Adv. Appl. 116, Birkhäuser, Basel, 2000. 10.1007/978-3-0348-8401-3Search in Google Scholar
[13] N. B. Hamadi and S. Omri, Uncertainty principles for the continuous wavelet transform in the Hankel setting, Appl. Anal. 97 (2018), no. 4, 513–527. 10.1080/00036811.2016.1276169Search in Google Scholar
[14]
A. Hassini and K. Trimèche,
Wavelets and generalized windowed transforms associated with the Dunkl–Bessel–Laplace operator on
[15] K. Hikami, Dunkl operator formalism for quantum many-body problems associated with classical root systems, J. Phys. Soc. Japan 65 (1996), no. 2, 394–401. 10.1143/JPSJ.65.394Search in Google Scholar
[16]
E. Malinnikova,
Orthonormal sequences in
[17] H. Mejjaoli, Generalized Dunkl–Sobolev spaces of exponential type and applications, JIPAM. J. Inequal. Pure Appl. Math. 10 (2009), no. 2, Paper No. 55. Search in Google Scholar
[18] 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
[19] H. Mejjaoli and N. Sraieb, Uncertainty principles for the Dunkl–Bessel transform, Math. Sci. Res. J. 15 (2011), no. 8, 245–263. Search in Google Scholar
[20]
H. Mejjaoli, N. Sraieb and K. Trimèche,
Inversion theorem and quantitative uncertainty principles for the Dunkl Gabor transform on
[21] H. Mejjaoli and K. Trimèche, Harmonic analysis associated with the Dunkl–Bessel Laplace operator and a mean value property, Fract. Calc. Appl. Anal. 4 (2001), no. 4, 443–480. Search in Google Scholar
[22] W. Nefzi, Weinstein multipliers of Laplace transform type, Integral Transforms Spec. Funct. 29 (2018), no. 6, 470–488. 10.1080/10652469.2018.1459601Search in Google Scholar
[23] W. Nefzi, Fractional integrals for the Weinstein operator, Integral Transforms Spec. Funct. 31 (2020), no. 11, 906–920. 10.1080/10652469.2020.1761803Search in Google Scholar
[24] S. Saitoh, Best approximation, Tikhonov regularization and reproducing kernels, Kodai Math. J. 28 (2005), no. 2, 359–367. 10.2996/kmj/1123767016Search in Google Scholar
[25] S. Saitoh, Theory of reproducing kernels: Applications to approximate solutions of bounded linear operator equations on Hilbert spaces, Selected Papers on Analysis and Differential Equations, Amer. Math. Soc. Transl. Ser. 2 230, American Mathematical Society, Providence (2010), 107–134. 10.1090/trans2/230/06Search in Google Scholar
[26] S. Saitoh and Y. Sawano, Theory of Reproducing Kernels and Applications, Dev. Math. 44, Springer, Singapore, 2016. 10.1007/978-981-10-0530-5Search in Google Scholar
[27]
H. S. Shapiro,
Uncertainty principles for basis in
[28] F. Soltani, Uncertainty principles for the Dunkl–Wigner transforms, J. Oper. 2016 (2016), Article ID 7637346. 10.1155/2016/7637346Search in Google Scholar
[29] F. Soltani, The Dunkl–Wigner transforms on the real line, J. Nonlinear Funct. Anal. 2017 (2017), Paper No. 24. 10.23952/jnfa.2017.24Search in Google Scholar
[30] F. Soltani, Reconstruction and best approximate inversion formulas for the modified Whittaker–Stockwell transform, Ramanujan J. 65 (2024), no. 1, 313–331. 10.1007/s11139-024-00900-ySearch in Google Scholar
[31] F. Soltani and I. Maktouf, Dunkl–Weinstein multiplier operators and applications to reproducing kernel theory, Mediterr. J. Math. 21 (2024), no. 3, Paper No. 80. 10.1007/s00009-024-02623-2Search in Google Scholar
[32] F. Soltani and I. Maktouf, Localization operators and inversion formulas for the Dunkl–Weinstein–Stockwell transform, Georgian Math. J. 31 (2024), no. 2, 339–354. 10.1515/gmj-2023-2077Search in Google Scholar
[33] F. Soltani, I. Maktouf and W. Nefzi, Heisenberg-type uncertainty principles for the Dunkl–Weinstein transform, Asian-Eur. J. Math. 16 (2023), no. 6, Article ID 2350102. 10.1142/S1793557123501024Search in Google Scholar
[34] F. Soltani and Y. Zarrougui, Localization operators and Shapiro’s inequality for the Sturm–Liouville–Stockwell transform, J. Math. Sci. (2024), 10.1007/s10958-024-07090-4. 10.1007/s10958-024-07090-4Search in Google Scholar
[35] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University, Cambridge, 1966. Search in Google Scholar
[36] A. Weinstein, Singular partial differential equations and their applications, Fluid Dynamics and Applied Mathematics, Gordon and Breach, New York (1962), 29–49. Search in Google Scholar
[37] L. M. Yang, A note on the quantum rule of the harmonic oscillator, Phys. Rev. (2) 84 (1951), 788–790. 10.1103/PhysRev.84.788Search in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston