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$L^p$ LOCAL UNCERTAINTY PRINCIPLE FOR THE DUNKL TRANSFORM

Year 2015, Volume: 3 Issue: 2, 100 - 109, 01.10.2015

Fethi Soltanı

Abstract

In this paper, we establish $L^p$ local uncertainty principle for the Dunkl transform on Rd; and we deduce $L^p$ version of the Heisenberg-Pauli- Weyl uncertainty principle for this transform. We use also the $L^p$ local uncertainty principle for the Dunkl transform and the techniques of Donoho-Stark, we obtain uncertainty principles of concentration type in the $L^p$ theory, when 1 < p < 2.

Keywords

Dunkl transform; local uncertainty principle; Heisenberg-Pauli-Weyl uncertainty principle; Donoho-Stark's uncertainty principles.

References

  • [1] M. Cowling and J.F. Price, Bandwidth versus time concentration: the Heisenberg-Pauli-Weyl inequality, SIAM J. Math. Anal. Vol:15 (1984), 151-165.
  • [2] D.L. Donoho and P.B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. Vol:49, No.3 (1989), 906-931.
  • [3] C.F. Dunkl, Integral kernels with re ection group invariance, Canad. J. Math. Vol:43 (1991), 1213-1227.
  • [4] C.F. Dunkl, Hankel transforms associated to nite re ection groups, Contemp. Math. Vol:138 (1992), 123-138.
  • [5] W.G. Faris, Inequalities and uncertainty inequalities, Math. Phys. Vol:19 (1978), 461-466.
  • [6] I.I. Hirschman, A note on entropy, Amer. J. Math. Vol:79 (1957), 152-156.
  • [7] M.F.E.de Jeu, The Dunkl transform, Invent. Math. Vol:113 (1993), 147-162.
  • [8] J.F. Price, Inequalities and local uncertainty principles, J. Math. Phys. Vol:24 (1983), 1711- 1714.
  • [9] J.F. Price, Sharp local uncertainty principles, Studia Math. Vol:85 (1987), 37-45.
  • [10] M. Rosler, An uncertainty principle for the Dunkl transform, Bull. Austral. Math. Soc. Vol:59 (1999), 353-360.
  • [11] N. Shimeno, A note on the uncertainty principle for the Dunkl transform, J. Math. Sci. Univ. Tokyo Vol:8 (2001), 33-42.
  • [12] F. Soltani, Heisenberg-Pauli-Weyl uncertainty inequality for the Dunkl transform on Rd, Bull. Austral. Math. Soc. Vol:87 (2013), 316-325.
  • [13] F. Soltani, A general form of Heisenberg-Pauli-Weyl uncertainty inequality for the Dunkl transform, Int. Trans. Spec. Funct. Vol:24, No.5 (2013), 401-409.
  • [14] F. Soltani, Donoho-Stark uncertainty principle associated with a singular secondorder di erential operator, Int. J. Anal. Appl. Vol:4, No.1 (2014), 1-10.
  • [15] F. Soltani, Lp uncertainty principles on Sturm-Liouville hypergroups, Acta Math. Hungar. Vol:142, No.2 (2014), 433-443.
  • [16] F. Soltani, Lp local uncertainty inequality for the Sturm-Liouville transform, CUBO Math. J. Vol:16, No.1 (2014), 95-104.
  • [17] F. Soltani, An Lp Heisenberg-Pauli-Weyl uncertainty principle for the Dunkl transform, Konuralp J. Math. Vol:2, No.1 (2014), 1-6.
  • [18] F. Soltani, Lp Donoho-Stark uncertainty principles for the Dunkl transform on Rd, J. Phys. Math. Vol:5, No.1 (2014), 4 pages.
  • [19] E.M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. Vol:83 (1956), 482-492.
  • [20] E.M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press., Princeton, N.J, 1971.

Year 2015, Volume: 3 Issue: 2, 100 - 109, 01.10.2015

Fethi Soltanı

Abstract

References

  • [1] M. Cowling and J.F. Price, Bandwidth versus time concentration: the Heisenberg-Pauli-Weyl inequality, SIAM J. Math. Anal. Vol:15 (1984), 151-165.
  • [2] D.L. Donoho and P.B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. Vol:49, No.3 (1989), 906-931.
  • [3] C.F. Dunkl, Integral kernels with re ection group invariance, Canad. J. Math. Vol:43 (1991), 1213-1227.
  • [4] C.F. Dunkl, Hankel transforms associated to nite re ection groups, Contemp. Math. Vol:138 (1992), 123-138.
  • [5] W.G. Faris, Inequalities and uncertainty inequalities, Math. Phys. Vol:19 (1978), 461-466.
  • [6] I.I. Hirschman, A note on entropy, Amer. J. Math. Vol:79 (1957), 152-156.
  • [7] M.F.E.de Jeu, The Dunkl transform, Invent. Math. Vol:113 (1993), 147-162.
  • [8] J.F. Price, Inequalities and local uncertainty principles, J. Math. Phys. Vol:24 (1983), 1711- 1714.
  • [9] J.F. Price, Sharp local uncertainty principles, Studia Math. Vol:85 (1987), 37-45.
  • [10] M. Rosler, An uncertainty principle for the Dunkl transform, Bull. Austral. Math. Soc. Vol:59 (1999), 353-360.
  • [11] N. Shimeno, A note on the uncertainty principle for the Dunkl transform, J. Math. Sci. Univ. Tokyo Vol:8 (2001), 33-42.
  • [12] F. Soltani, Heisenberg-Pauli-Weyl uncertainty inequality for the Dunkl transform on Rd, Bull. Austral. Math. Soc. Vol:87 (2013), 316-325.
  • [13] F. Soltani, A general form of Heisenberg-Pauli-Weyl uncertainty inequality for the Dunkl transform, Int. Trans. Spec. Funct. Vol:24, No.5 (2013), 401-409.
  • [14] F. Soltani, Donoho-Stark uncertainty principle associated with a singular secondorder di erential operator, Int. J. Anal. Appl. Vol:4, No.1 (2014), 1-10.
  • [15] F. Soltani, Lp uncertainty principles on Sturm-Liouville hypergroups, Acta Math. Hungar. Vol:142, No.2 (2014), 433-443.
  • [16] F. Soltani, Lp local uncertainty inequality for the Sturm-Liouville transform, CUBO Math. J. Vol:16, No.1 (2014), 95-104.
  • [17] F. Soltani, An Lp Heisenberg-Pauli-Weyl uncertainty principle for the Dunkl transform, Konuralp J. Math. Vol:2, No.1 (2014), 1-6.
  • [18] F. Soltani, Lp Donoho-Stark uncertainty principles for the Dunkl transform on Rd, J. Phys. Math. Vol:5, No.1 (2014), 4 pages.
  • [19] E.M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. Vol:83 (1956), 482-492.
  • [20] E.M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press., Princeton, N.J, 1971.
There are 20 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Fethi Soltanı

Publication Date October 1, 2015
Submission Date July 10, 2014
Published in Issue Year 2015 Volume: 3 Issue: 2

Cite

APA Soltanı, F. (2015). $L^p$ LOCAL UNCERTAINTY PRINCIPLE FOR THE DUNKL TRANSFORM. Konuralp Journal of Mathematics, 3(2), 100-109.
AMA Soltanı F. $L^p$ LOCAL UNCERTAINTY PRINCIPLE FOR THE DUNKL TRANSFORM. Konuralp J. Math. October 2015;3(2):100-109.
Chicago Soltanı, Fethi. “$L^p$ LOCAL UNCERTAINTY PRINCIPLE FOR THE DUNKL TRANSFORM”. Konuralp Journal of Mathematics 3, no. 2 (October 2015): 100-109.
EndNote Soltanı F (October 1, 2015) $L^p$ LOCAL UNCERTAINTY PRINCIPLE FOR THE DUNKL TRANSFORM. Konuralp Journal of Mathematics 3 2 100–109.
IEEE F. Soltanı, “$L^p$ LOCAL UNCERTAINTY PRINCIPLE FOR THE DUNKL TRANSFORM”, Konuralp J. Math., vol. 3, no. 2, pp. 100–109, 2015.
ISNAD Soltanı, Fethi. “$L^p$ LOCAL UNCERTAINTY PRINCIPLE FOR THE DUNKL TRANSFORM”. Konuralp Journal of Mathematics 3/2 (October 2015), 100-109.
JAMA Soltanı F. $L^p$ LOCAL UNCERTAINTY PRINCIPLE FOR THE DUNKL TRANSFORM. Konuralp J. Math. 2015;3:100–109.
MLA Soltanı, Fethi. “$L^p$ LOCAL UNCERTAINTY PRINCIPLE FOR THE DUNKL TRANSFORM”. Konuralp Journal of Mathematics, vol. 3, no. 2, 2015, pp. 100-9.
Vancouver Soltanı F. $L^p$ LOCAL UNCERTAINTY PRINCIPLE FOR THE DUNKL TRANSFORM. Konuralp J. Math. 2015;3(2):100-9.
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