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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Paley-Wiener theorems for the Dunkl transform


Author: Marcel de Jeu
Journal: Trans. Amer. Math. Soc. 358 (2006), 4225-4250
MSC (2000): Primary 33C52; Secondary 43A32, 33C80, 22E30
Published electronically: May 9, 2006
MathSciNet review: 2231377
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Abstract: We conjecture a geometrical form of the Paley-Wiener theorem for the Dunkl transform and prove three instances thereof, by using a reduction to the one-dimensional even case, shift operators, and a limit transition from Opdam's results for the graded Hecke algebra, respectively. These Paley-Wiener theorems are used to extend Dunkl's intertwining operator to arbitrary smooth functions.

Furthermore, the connection between Dunkl operators and the Cartan motion group is established. It is shown how the algebra of radial parts of invariant differential operators can be described explicitly in terms of Dunkl operators. This description implies that the generalized Bessel functions coincide with the spherical functions. In this context of the Cartan motion group, the restriction of Dunkl's intertwining operator to the invariants can be interpreted in terms of the Abel transform. We also show that, for certain values of the multiplicities of the restricted roots, the Abel transform is essentially inverted by a differential operator.


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Additional Information

Marcel de Jeu
Affiliation: Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands
Email: mdejeu@math.leidenuniv.nl

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03960-2
PII: S 0002-9947(06)03960-2
Keywords: Dunkl operator, Paley--Wiener theorem, graded Hecke algebra, Cartan motion group, spherical function, multivariable Bessel function
Received by editor(s): April 23, 2004
Published electronically: May 9, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.