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An uncertainty principle for the Dunkl transform. (English) Zbl 0939.33012

The affine Dunkl transform on \(\mathbb{R}^N\) is an integral transform which generalizes the classical Fourier transform; it appears in the study of certain differential-difference operators (called Dunkl operators) which play a major role in the algebraic description of quantum many body systems of Calogero-Moser-Sutherland type.
The paper under review contains a Heisenberg-Weyl uncertainty principle for this Dunkl transform. The proof uses properties of multivariate Hermite polynomials associated with Dunkl operators (which are due to T. H. Baker, P. J. Forrester, and the author) and is based on old ideas of N. G. de Bruijn and C. T. Roosenraad for the classical Fourier transform. It should be mentioned that in the one-dimensional case a stronger version of the Heisenberg-Weyl inequality for nonsymmetric functions was given by the author and M. Voit [Proc. Am. Math. Soc. 127, 183-194 (1999; Zbl 0910.44003)].

MSC:

33C80 Connections of hypergeometric functions with groups and algebras, and related topics
43A32 Other transforms and operators of Fourier type
42C15 General harmonic expansions, frames
26D10 Inequalities involving derivatives and differential and integral operators

Citations:

Zbl 0910.44003
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References:

[1] DOI: 10.1090/S0002-9939-99-04553-0 · Zbl 0910.44003 · doi:10.1090/S0002-9939-99-04553-0
[2] Rösier, Duke Math. J.
[3] DOI: 10.1007/s002200050307 · Zbl 0908.33005 · doi:10.1007/s002200050307
[4] DOI: 10.1103/PhysRevLett.69.703 · Zbl 0968.37521 · doi:10.1103/PhysRevLett.69.703
[5] DOI: 10.1007/BF02099456 · Zbl 0859.35103 · doi:10.1007/BF02099456
[6] Opdam, Compositio Math. 85 pp 333– (1993)
[7] Bruijn, Inequalities pp 57– (1967)
[8] DOI: 10.1007/BF01244305 · Zbl 0789.33007 · doi:10.1007/BF01244305
[9] Dunkl, Canad. J. Math. 43 pp 1213– (1991) · Zbl 0827.33010 · doi:10.4153/CJM-1991-069-8
[10] DOI: 10.2307/2001022 · Zbl 0652.33004 · doi:10.2307/2001022
[11] DOI: 10.1215/S0012-7094-98-09501-1 · Zbl 0948.33012 · doi:10.1215/S0012-7094-98-09501-1
[12] DOI: 10.1137/0149053 · Zbl 0689.42001 · doi:10.1137/0149053
[13] Chihara, An introduction to orthogonal polynomials (1978) · Zbl 0389.33008
[14] DOI: 10.1006/jfan.1994.1067 · Zbl 0802.47051 · doi:10.1006/jfan.1994.1067
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