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A Variation on Uncertainty Principles for Quaternion Linear Canonical Transform

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Abstract

In this paper we prove Clarkson-type and Nash-type inequalities in the (right-sided) Quaternion Linear Canonical transform (QLCT) for \(L^p\)-functions. Next, we show Heisenberg-type inequalities and Matolcsi–Szücs-type inequality for the QLCT. Finally, we deduce local-type uncertainty inequalities for the Quaternion Linear Canonical transform.

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References

  1. Abe, S., Sheridan, J.T.: Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation. Opt. Lett. 19, 1801–1803 (1994)

    Article  ADS  Google Scholar 

  2. Achak, A., Abouelaz, A., Daher, R., Safouane, N.: Uncertainty principles for the quaternion linear canonical transform. Adv. Appl. Clifford Algebras. 29(99), 1–19 (2019)

    MathSciNet  MATH  Google Scholar 

  3. Bahri, M., Ashino, R.: A simplified proof of uncertainty principle for quaternion linear canonical transform. Abstr. Appl. Anal. (2016). https://doi.org/10.1155/2016/5874930. (Article ID 5874930)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bahri, M., Ashino, R.: A variation on uncertainty principle and logarithmic uncertainty principle for continuous quaternion wavelet transforms. Abstr. Appl. Anal. (2017). https://doi.org/10.1155/2017/3795120. (Article ID 3795120)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bahri, M., Resnawati, M.S.: A version of uncertainty principle for quaternion linear canonical transform. Abstr. Appl. Anal. (2018). https://doi.org/10.1155/2018/8732457. (Article ID 8732457)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bahri, M., Hitzer, E.S., Hayashi, A., Ashino, R.: An uncertainty principle for quaternion Fourier transform. Comput. Math. Appl. 56(9), 2398–2410 (2008)

    Article  MathSciNet  Google Scholar 

  7. Barshan, B., Kutay, M., Ozaktas, H.: Optimal filtering with linear canonical transformations. Opt. Commun. 135, 32–36 (1997)

    Article  ADS  Google Scholar 

  8. Bernardo, L.M.: ABCD matrix formalism of fractional Fourier optics. Opt. Eng. (Bellingham) 35, 732–740 (1996)

    Article  ADS  Google Scholar 

  9. Collins, S.A.J.: Lens-system diffraction integral written in terms of matrix optics. J. Opt. Soc. Am. 60(9), 1168–1177 (1970)

    Article  ADS  Google Scholar 

  10. Cowling, M., Price, J.F.: Bandwidth versus time concentration: the Heisenberg–Pauli–Weyl inequality. SIAM J. Math. Anal. 15, 151–165 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  11. El Haoui, Y., Hitzer, E.: Generalized uncertainty principles associated with the quaternionic offset linear canonical transform. Complex Var. Elliptic Equ. (2021). https://doi.org/10.1080/17476933.2021.1916919

    Article  Google Scholar 

  12. Ghazouani, S., Soltani, E.A., Fitouhi, A.: A unified class of integral transforms related to the Dunkl transform. J. Math. Anal. Appl. 449(2), 1797–1849 (2017)

    Article  MathSciNet  Google Scholar 

  13. Ghobber, S.: Uncertainty principles involving \(L^1\)-norms for the Dunkl transform. Int. Trans. Spec. Funct. 24(6), 491–501 (2013)

    Article  MathSciNet  Google Scholar 

  14. Ghobber, S.: Variations on uncertainty principles for integral operators. Appl. Anal. 93(5), 1057–1072 (2014)

    Article  MathSciNet  Google Scholar 

  15. Ghobber, S., Jaming, P.: Uncertainty principles for integral operators. Stud. Math. 220(3), 197–220 (2014)

    Article  MathSciNet  Google Scholar 

  16. Hitzer, E.: Quaternion Fourier transform on quaternion fields and generalizations. Adv. Appl. Clifford Algebra 17(3), 497–517 (2007)

    Article  MathSciNet  Google Scholar 

  17. Hleili, K.: Uncertainty principles for spherical mean \(L^2\)-multiplier operators. J. Pseudodiffer. Oper. Appl. 9, 573–587 (2018)

    Article  MathSciNet  Google Scholar 

  18. Hleili, K.: Continuous wavelet transform and uncertainty principle related to the Weinstein operator. Integral Transf. Spec. Funct. 29(4), 252–268 (2018)

    Article  MathSciNet  Google Scholar 

  19. Hleili, K.: Some results for the windowed Fourier transform related to the spherical mean operator. Acta Math. Vietnam. 46, 179–201 (2021)

    Article  MathSciNet  Google Scholar 

  20. Hu, B., Zhou, Y., Lie, L.D., Zhang, J.Y.: Polar linear canonical transforming quaternion domain. J. Inf. Hiding Multimed. Signal Process. 6(6), 1185–1193 (2015)

    Google Scholar 

  21. James, D.F.V., Agarwal, G.S.: The generalized Fresnel transform and its application to optics. Opt. Commun. 126, 207–212 (1996)

    Article  ADS  Google Scholar 

  22. Kerr, F.H.: A fractional power theory for Hankel transforms in \(L^2(\mathbb{R}_+)\). J. Math. Anal. Appl. 158(1), 114–123 (1991)

    Article  MathSciNet  Google Scholar 

  23. Kou, K.I., Morais, J.: Asymptotic behaviour of the quaternion linear canonical transform and the Bochner–Minlos theorem. Appl. Math. Comput. 247(15), 675–688 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Kou, K.I., Ou, J.Y., Morais, J.: On uncertainty principle for quaternionic linear canonical transform. Abstr. Appl. Anal. 2013, 725952 (2013)

    Article  MathSciNet  Google Scholar 

  25. Laeng, E., Morpurgo, C.: An uncertainty inequality involving \(L^1\)-norms. Proc. Am. Math. Soc. 127(12), 3565–3572 (1999)

    Article  Google Scholar 

  26. Lian, P.: Uncertainty principle for the quaternion Fourier transform. J. Math. Anal. Appl. 467(2), 1258–1269 (2018). https://doi.org/10.1016/j.jmaa.2018.08.002. (MR 3842432)

    Article  MathSciNet  MATH  Google Scholar 

  27. Li-Ping, C., Kit Ian Kou, K., Ming-Sheng, L.: Pitt’s inequality and the uncertainty principle associated with the quaternion Fourier transform. J. Math. Anal. Appl. 423, 681–700 (2015)

    Article  MathSciNet  Google Scholar 

  28. Morpurgo, C.: Extremals of some uncertainty inequalities. Bull. Lond. Math. Soc. 33(1), 52–58 (2001)

    Article  MathSciNet  Google Scholar 

  29. Moshinsky, M., Quesne, C.: Linear canonical transformations and their unitary representations. J. Math. Phys. 12(8), 1772–1780 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  30. Ozaktas, H., Zalevsky, Z., Kutay, M.: The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, New York (2001)

    Google Scholar 

  31. Pei, S., Ding, J.: Eigenfunctions of linear canonical transform. IEEE Trans. Signal Process. 50, 11–26 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  32. Price, J.F.: Inequalities and local uncertainty principles. J. Math. Phys. 24, 1711–1714 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  33. Soltani, F.: An \(L^p\) Heisenberg–Pauli–Weyl uncertainty principle for the Dunkl transform. Konuralp J. Math. (2) 1, 1–6 (2014)

    MATH  Google Scholar 

  34. Soltani, F.: \(L^p\) local uncertainty principle for the Dunkl transform. Konuralp J. Math. 3(2), 100–109 (2015)

    MathSciNet  MATH  Google Scholar 

  35. Wolf, K.B.: Canonical transforms II. Complex radial transforms. J. Math. Phys. 15, 1295–1301 (1974)

    Article  ADS  MathSciNet  Google Scholar 

  36. Yang, Y., Kou, K.I.: Uncertainty principles for hypercomplex signals in the linear canonical transform domains. Signal Process. 95, 67–75 (2014)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Preparatory Institute for Engineering Studies of Kairouan, Kairouan University, Kairouan, Tunisia

    Khaled Hleili

  2. Department of Mathematics, Faculty of Science, Northern Borders University, Arar, Saudi Arabia

    Khaled Hleili

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  1. Khaled Hleili
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Correspondence to Khaled Hleili.

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Communicated by Eckhard Hitzer.

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Hleili, K. A Variation on Uncertainty Principles for Quaternion Linear Canonical Transform. Adv. Appl. Clifford Algebras 31, 46 (2021). https://doi.org/10.1007/s00006-021-01147-2

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