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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])
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Abstract

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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