Paley–Wiener theorems for the Dunkl transform

Abstract

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 PaleyWiener 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.

Dunkl spectral multipliers with values in UMD lattices

“…and this formula coincides for particular root systems and particular values of κ to the radial part of the Laplace-Beltrami operator of a Riemannian symmetric space of Euclidean type (see [29]). More generally, Dunkl operators have significantly contributed to the development of harmonic analysis associated with a root system and to the theory of multivariable hypergeometric functions.…”

The Range of the Spectral Projection Associated with the Dunkl Laplacian

“…for some positive constant C N ; see for instance [34]. This result has been generalized by de Jeu [23] to the Dunkl transform. To state the (complex) Paley-Wiener theorem for F k , we introduce the following notation.…”

“…As an application of the main result, we prove a spectral version of the complex Paley-Wiener theorem for the Dunkl transform F k given in [23]. More precisely, we characterize the set of functions φðs, ηÞ defined on ℝ × S d−1 for which there exists a compactly supported smooth function f with support in BðO, RÞ so that φðs, ηÞ = F k ð f ÞðsηÞ (see Theorem 10).…”

“…Fact 4 (see [23]). The Dunkl transform F k is a linear isomorphism between D R ðℝ d Þ and H R ðℂ d Þ, for all R > 0:…”

A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian

“…We next turn to the Bessel functions which will show up in the Mehler-Heine formula. They are given in terms of Bessel functions of Dunkl type which generalize the spherical functions of Cartan motion groups; see [6] and [21] for a general background. We denote by J B k the Bessel function which is associated with the rational Dunkl operators for the root system B q = {±e i , ±e i ± e j : 1 ≤ i < j ≤ q} and multiplicity k = (k 1 , k 2 ) corresponding to the roots ±e i and ±e i ± e j .…”