Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-14T09:29:31.201Z Has data issue: false hasContentIssue false
  • English
  • Français

HEISENBERG–PAULI–WEYL UNCERTAINTY INEQUALITY FOR THE DUNKL TRANSFORM ON ℝd

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  19 October 2012

FETHI SOLTANI
FETHI SOLTANI*
Affiliation:
Higher College of Technology and Informatics, Street of the Employers 45, Charguia 2, 2035 Tunis, Tunisia (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we give analogues of the local uncertainty inequality for the Dunkl transform on ℝd, and indicate how the local uncertainty inequality implies a global uncertainty inequality.

Keywords

Dunkl transformlocal uncertainty principleglobal uncertainty principleHeisenberg–Pauli–Weyl uncertainty principle

MSC classification

Secondary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 2 , April 2013 , pp. 316 - 325
Copyright
©2012 Australian Mathematical Publishing Association Inc.

References

[1]Dunkl, C. F., ‘Integral kernels with reflection group invariance’, Canad. J. Math. 43 (1991), 1213–1227.CrossRefGoogle Scholar
[2]Dunkl, C. F., ‘Hankel transforms associated to finite reflection groups’, Contemp. Math. 138 (1992), 123–138.CrossRefGoogle Scholar
[3]Faris, W. G., ‘Inequalities and uncertainty inequalities’, J. Math. Phys. 19 (1978), 461–466.CrossRefGoogle Scholar
[4]Folland, G. B. & Sitaram, A., ‘The uncertainty principle: a mathematical survey’, J. Fourier Anal. Appl. 3 (1997), 207–238.CrossRefGoogle Scholar
[5]Havin, V. & Jöricke, B., The Uncertainty Principle in Harmonic Analysis (Springer, Berlin, 1994).CrossRefGoogle Scholar
[6]Heisenberg, W., ‘Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik’, Z. Phys. A 43 (1927), 172–198.CrossRefGoogle Scholar
[7]de Jeu, M. F. E., ‘The Dunkl transform’, Invent. Math. 113 (1993), 147–162.CrossRefGoogle Scholar
[8]Price, J. F., ‘Inequalities and local uncertainty principles’, J. Math. Phys. 24 (1983), 1711–1714.CrossRefGoogle Scholar
[9]Price, J. F., ‘Sharp local uncertainty principles’, Studia Math. 85 (1987), 37–45.CrossRefGoogle Scholar
[10]Rösler, M., ‘Positivity of Dunkl’s intertwining operator’, Duke Math. J. 98 (1999), 445–463.CrossRefGoogle Scholar
[11]Rösler, M., ‘An uncertainty principle for the Dunkl transform’, Bull. Austral. Math. Soc. 59 (1999), 353–360.CrossRefGoogle Scholar
[12]Shimeno, N., ‘A note on the uncertainty principle for the Dunkl transform’, J. Math. Sci. Univ. Tokyo 8 (2001), 33–42.Google Scholar
[13]Watson, G. N., A Treatise on Theory of Bessel Functions (Cambridge University Press, Cambridge, 1966).Google Scholar
[14]Weyl, H., Gruppentheorie und Quantenmechanik (S. Hirzel, Leipzig, 1928). Revised English edition: The Theory of Groups and Quantum Mechanics (Methuen, London, 1931); (reprinted by Dover, New York, 1950).Google Scholar