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Harmonic analysis problems associated with the k-Hankel Gabor transform

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Abstract

We introduce a continuous k-Hankel Gabor transform acting on a Hilbert space deforming \(L^2(\mathbb R)\). We prove a Plancherel formula and \(L^2\)-inversion formulas for it. We also prove several uncertainty principles for this transform such as Heisenberg type inequalities and Faris–Price type uncertainty principle.

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References

  1. Abreu, L.D.: Sampling and interpolation in Bargmann–Fock spaces of poly-analytic functions. Appl. Comput. Harmon. Anal. 29, 287–302 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Ben Saïd, S., Kobayashi, T., Ørsted, B.: Laguerre semigroup and \((k, a)\)-generalized operators. Compos. Math. 148(04), 1265–1336 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Ben Saïd, S.: Strichartz estimates for Schrödinger–Laguerre operators. Semigroup Forum 90, 251–269 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Ben Saïd, S.: A product formula and a convolution structure for a \(k\)-Hankel transform on \(\mathbb{R}\). J. Math. Anal. Appl. 463(2), 1132–1146 (2018)

    MathSciNet  MATH  Google Scholar 

  5. Ben Saïd, S., Deleaval, L.: A Hardy-Littlewood maximal operator for the generalized Fourier transform on \({\mathbb{R}}\). J. Geom. Anal. (2019). https://doi.org/10.1007/s12220-019-00183-6

    Article  MATH  Google Scholar 

  6. Ben Salem, N., Nasr, A.R.: Shapiro type inequalities for the Weinstein and the Weinstein Gabor transforms. Konuralp J. Math. 5(1), 68–76 (2017)

    MathSciNet  MATH  Google Scholar 

  7. Bensal, A., Kumar, A.: Heisenberg uncertainty inequality for Gabor transform. J. Math. Inequal. 10(3), 737–749 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Bansal, A., Kumar, A.: Qualitative uncertainty principle for Gabor transform. Bull. Korean Math. Soc. 54(1), 71–84 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Czaja, W., Gigante, G.: Continuous Gabor transform for strong hypergroups. J Fourier Anal Appl. 9, 321–339 (2003)

    MathSciNet  MATH  Google Scholar 

  10. Ciatti, P., Ricci, F., Sundari, M.: Heisenberg–Pauli–Weyl uncertainty inequalities and polynomial volume growth. Adv. Math. 215, 616–625 (2007)

    MathSciNet  MATH  Google Scholar 

  11. Cohen, L.: Generalized phase-space distribution functions. J. Math. Phys. 7, 781–786 (1966)

    MathSciNet  Google Scholar 

  12. Cohen, L.: Time-frequency distributions—a review. Proc. IEEE 77, 941–981 (1989)

    Google Scholar 

  13. De Bie, H., Lian, P., Constales, D.: Explicit formulas for the Dunkl dihedral kernel and the \((\kappa, a)\)-generalized Fourier kernel. J. Math. Anal. Appl. 460(2), 900–926 (2018)

    MathSciNet  MATH  Google Scholar 

  14. De Bie, H., Oste, R., Van der Jeugt, J.: Generalized Fourier transforms arising from the enveloping algebras of \((2)\) and \({\mathfrak{o}}{\mathfrak{s}}{\mathfrak{p}}(1|2)\). Int. Math. Res. Not. 15, 4649–4705 (2016)

    MathSciNet  MATH  Google Scholar 

  15. De Bie, H., Ørsted, B., Somberg, P., Souček, V.: \(k\)-Hankel operators and a family of realizations of osp(1|2). Trans. Am. Math. Soc. 364, 3875–3902 (2012)

    MATH  Google Scholar 

  16. Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311, 167–183 (1989)

    MathSciNet  MATH  Google Scholar 

  17. Dunkl, C.F.: Hankel transforms associated to finite reflection groups. In: Proceedings of the Special Session on Hypergeometric Functions on Domains of Positivity, Jack polynomials and applications. (Tampa, FL, 1991), Contemp. Math., 1992; vol. 138, pp. 123–138 (1992)

  18. Debnath, L.: Wavelet Transforms and Their Applications. Birkhäuser, Boston (2002)

    MATH  Google Scholar 

  19. Daubechies, Time-frequency: localization operators: a geometric phase space approach. IEEE Trans. Inform. Theory 34, 605–612 (1988)

    MathSciNet  MATH  Google Scholar 

  20. Faris, W.G.: Inequalities and uncertainty inequalities. Math. Phys. 19, 461–466 (1978)

    Google Scholar 

  21. Folland, G.-B., Sitaram, A.: The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3(3), 207–238 (1997)

    MathSciNet  MATH  Google Scholar 

  22. Fernandez, C., Galbis, A.: Annihilating sets for the short time Fourier transform. Adv. Math. 224(5), 1904–1926 (2010)

    MathSciNet  MATH  Google Scholar 

  23. Fernandez, C., Galbis, A., Martinez, J.: Localization operators and an uncertainty principle for the discrete short time Fourier transform. Abstr. Appl. Anal. 2014, Art. ID 131459 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Gabor, D.: Theory of communication. Part 1: The analysis of information. J. Inst. Electr. Eng. Part III Radio Commun. Eng. 93(26), 429–441 (1946)

    Google Scholar 

  25. Gröchenig, K.: Aspects of Gabor analysis on locally compact abelian groups. Gabor analysis and algorithms, pp. 211–231, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA (1998)

  26. Ghobber, S., Omri, S.: Time–frequency concentration of the windowed Hankel transform. Integral Transf. Spec. Funct. 25(6), 481–496 (2014)

    MathSciNet  MATH  Google Scholar 

  27. Ghobber, S., Jaming, P.: Uncertainty principles for integral orperators. Stud. Math. 220, 197–220 (2014)

    MATH  Google Scholar 

  28. Havin, V., Jöricke, B.: The Uncertainty Principle in Harmonic Analysis. Springer, Berlin (1994)

    MATH  Google Scholar 

  29. Hutnikova, M., Hutnik, O.: An alternative description of Gabor spaces and Gabor–Toeplitz operators. Rep. Math. Phys. 66(2), 237–250 (2010)

    MathSciNet  MATH  Google Scholar 

  30. Hamadi, N., Omri, S.: Uncertainty principles for the continuous wavelet transform in the Hankel setting. Appl. Anal. 97(4), 513–527 (2018)

    MathSciNet  Google Scholar 

  31. Gorbachev, D., Ivanov, V., Tikhonov, S.: Pitt’s inequalities and uncertainty principle for generalized Fourier transform. Int. Math. Res. Not. 2016(23), 7179–7200 (2016)

    MathSciNet  MATH  Google Scholar 

  32. Howe, R.: The oscillator semigroup. The mathematical heritage of Hermann Weyl (Durham, NC, 1987), 61-132, Proc. Sympos. Pure Math., 48, Amer. Math. Soc., Providence, RI (1988)

  33. Johansen, T.-R.: Weighted inequalities and uncertainty principles for the \((k, a)\)-generalized Fourier transform. Int. J. Math. 27(3), 1650019 (2016)

    MathSciNet  MATH  Google Scholar 

  34. Kobayashi, T., Mano, G.: The Schrödinger model for the minimal representation of the indefinite orthogonal group \(O(p,q)\). vi + 132pp. the Mem. Amer. Math. Soc. 212(1000) (2011)

  35. Malinnikova, E.: Orthonormal sequences in \(L^{2} ({\mathbb{R}}^{d})\) and time frequency localization. J. Fourier Anal. Appl. 16(6), 983–1006 (2010)

    MathSciNet  MATH  Google Scholar 

  36. Mejjaoli, H., Sraieb, N.: Uncertainty principles for the continuous Dunkl Gabor transform and the Dunkl continuous Gabor transform. Mediterr. J. Math. 5, 443–466 (2008)

    MathSciNet  MATH  Google Scholar 

  37. Mejjaoli, H., Sraieb, N.: Gabor transform in quantum calculus and applications. Fract. Calc. Appl. Anal. 12(3), 319–336 (2009)

    MathSciNet  MATH  Google Scholar 

  38. Mejjaoli, H., Ahmed Salem, A.-O.: Weinstein Gabor transform and applications. Adv. Pure Math. 2, 203–210 (2012)

    MATH  Google Scholar 

  39. Mejjaoli, H., Jelassi, M., Othmani, Y.: Multivariable Bessel Gabor transform and applications. Oper. Matrices 9(3), 637–657 (2015)

    MathSciNet  MATH  Google Scholar 

  40. Mejjaoli, H.: Spectral theorems associated with the \((k, a)\)-generalized wavelet multipliers. J. Pseudo-Differ. Oper. Appl. 9, 735–762 (2018)

    MathSciNet  MATH  Google Scholar 

  41. Mejjaoli, H., Trimèche, K.: k-Hankel two-wavelet theory and localization operators. Integral Transf. Spec. Funct. (2020). https://doi.org/10.1080/10652469.2020.1723011

    Article  MathSciNet  MATH  Google Scholar 

  42. Price, J.F.: Inequalities and local uncertainty principles. Math. Phys. 24, 1711–1714 (1978)

    MathSciNet  MATH  Google Scholar 

  43. Price, J.F.: Sharp local uncertainty principles. Stud. Math. 85, 37–45 (1987)

    MATH  Google Scholar 

  44. Price, J.F., Sitaram, A.: Local uncertainty inequalities for locally compact groups. Trans. Am. Math. Soc. 308, 105–114 (1988)

    MathSciNet  MATH  Google Scholar 

  45. Shannon, C.E.: A mathematical theory of communication. ACM SIGMOBILE Mobile Comput. Commun. Rev. 5(1), 3–55 (2001)

    MathSciNet  Google Scholar 

  46. Trimèche, K.: Generalized Wavelets and Hypergroups. Gordon and Breach Science, Amsterdam (1997)

    MATH  Google Scholar 

  47. Wilczok, E.: New uncertainty principles for the continuous Gabor transform and the continuous Gabor transform. Doc. Math. J. DMV 5, 201–226 (2000). (electronic)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are deeply indebted to the referees for providing constructive comments and helps in improving the contents of this article. The first author thanks professors K. Trimèche and M.W. Wong for their help.

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Authors and Affiliations

  1. College of Sciences, Department of Mathematics, Taibah University, PO BOX 30002, Al Madinah AL Munawarah, Saudi Arabia

    Hatem Mejjaoli

  2. Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain, Abu Dhabi, UAE

    Salem Ben Saïd

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  1. Hatem Mejjaoli
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  2. Salem Ben Saïd
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Correspondence to Hatem Mejjaoli.

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Mejjaoli, H., Ben Saïd, S. Harmonic analysis problems associated with the k-Hankel Gabor transform. J. Pseudo-Differ. Oper. Appl. 11, 1549–1593 (2020). https://doi.org/10.1007/s11868-020-00356-w

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