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The Dunkl transform. (English) Zbl 0789.33007

In the late eighties, Dunkel found a remarkable set of commuting operators that can be associated with a finite real reflection group. The operators contain complex parameters; if these parameters are all zero, Dunkl’s operators reduce to the ordinary directional derivatives. The operators have been studied by Dunkl, who obtained (amongst others) fairly detailed information about their action on polynomials. The present paper is concerned with the spectral problem for the Dunkl operators in the case that the real parts of the parameters are all nonnegative. We obtain estimates for the simultaneous eigenfunctions of the Dunkl operators and prove an inversion theorem and Plancherel formula for the associated integral transform. Plancherel-type results were obtained earlier by Dunkl, who exhibited an orthonormal basis for the \(L_ 2\)-space involved that consists of eigenfunctions for the transform with eigenvalues in \(\{\pm 1,\pm i\}\). Our method of proof is different and is almost exclusively based on exploiting the formal properties of the transform. The Dunkl transform contains Fourier analysis and the harmonic analysis for the Cartan motion group as a special case. This connection is explained in the paper (without proof).
Reviewer: M.F.E.de Jeu

MSC:

33C80 Connections of hypergeometric functions with groups and algebras, and related topics
44A15 Special integral transforms (Legendre, Hilbert, etc.)
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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References:

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