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An uncertainty principle for the Dunkl transform

Published online by Cambridge University Press:  17 April 2009

Margit Rösler
Margit Rösler
Affiliation:
Zentrum MathematikTechnische Universität MünchenArcisstr. 21, D-80290 München, Germany e-mail: [email protected]
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Abstract

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This note presents an analogue of the classical Heisenberg-Weyl uncertainty principle for the Dunkl transform on ℝN. Its proof is based on expansions with respect to generalised Hermite functions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 59 , Issue 3 , June 1999 , pp. 353 - 360
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Baker, T.H. and Forrester, P.J., ‘Non-symmetric Jack polynomials and integral kernels’, Duke Math. J. 95 (1998), 1–50.CrossRefGoogle Scholar
[2]Bruijn, N.G de, ‘Uncertainty principles in Fourier analysis’, in Inequalities, (Shisha, O., Editor) (Academic Press, New York, 1967), pp. 57–71.Google Scholar
[3]Chihara, T.S., An introduction to orthogonal polynomials, Mathematics and its Applications (Gordon and Breach, New York, 1978).Google Scholar
[4]Donoho, D.L. and Stark, P.B., ‘Uncertainty principle and signal recovery’, SIAM J. Appl. Math. 49 (1989), 906–931.Google Scholar
[5]Dunkl, C.F., ‘Differential-difference operators associated to reflection groups’, Trans. Amer. Math. Soc. 311 (1989), 167–183.CrossRefGoogle Scholar
[6]Dunkl, C.F., ‘Integral kernels with reflection group invariance’, Canad. J. Math. 43 (1991), 1213–1227.Google Scholar
[7]Jeu, M.F.E. de, ‘The Dunkl transform.’, Invent. Math. 113 (1993), 147–162.CrossRefGoogle Scholar
[8]Jeu, M.F.E. de, ‘An uncertainty principle for integral operators’, J. Funct. Anal. 122 (1994), 247–253.Google Scholar
[9]Lapointe, L. and Vinet, L., ‘Exact operator solution of the Calogero-Sutherland model’, Comm. Math. Phys. 178 (1996), 425–452.CrossRefGoogle Scholar
[10]Opdam, E.M., ‘Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group’, Compositio Math. 85 (1993), 333–373.Google Scholar
[11]Polychronakos, A.P., ‘Exchange operator formalism for integrable systems of particles’, Phys. Rev. Lett. 69 (1992), 703–705.CrossRefGoogle ScholarPubMed
[12]Rosier, M., ‘Generalized Hermite polynomials and the heat equation for Dunkl operators’, Comm. Math. Phys. 192 (1998), 519–542.Google Scholar
[13]Rösier, M., ‘Positivity of Dunkl's intertwining operator’, Duke Math. J. (to appear).Google Scholar
[14]Rosier, M. and Voit, M., ‘An uncertainty principle for Hankel transforms’, Proc. Amer. Math. Soc. 127 (1999), 183–194.Google Scholar
[15]Roosenraad, C.T., Inequalities with orthogonal polynomials, thesis (Univ. of Wisconsin, 1969).Google Scholar