Paley-Wiener theorems for the Dunkl transform
By Marcel de Jeu. 2004
116 citations recorded by the source corpus.
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 mo- tion group is established. It is shown how the algebra of radial parts of in- variant differential operators can be described explicitly in terms of Dunkl operators. This description implies that the generalized Bessel functions coin- cide with the spherical functions. In this context of the Cartan motion group, the restriction of Dunkl's intertwining operator to the invariants can be inter- preted 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. 1. Introduction and overview In recent times the study of special functions associated with root systems has developed to a considerable degree. Starting with a number of conjectures by Mac- donald, and the work of Heckman and Opdam on multivariable hypergeometric functions in the late 1980's, the development of the theory was greatly enhanced by the introduction of rational Dunkl operators by Dunkl (6). Through various intermediate steps of generalization, these operators can even be said to have ulti- mately provided crucial building blocks for Cherednik's work on double affine Hecke algebras and the q-Macdonald conjectures. Originally, before the introduction of Dunkl operators, the idea when studying special functions related to root systems was to consider root multiplicities in the theory of spherical functions on Lie groups as parameters, and then to develop a theory for Weyl group invariant objects in this more general situation, without the aid of the presence of the group. It was this point of view which underlay the Macdonald conjectures and which led Heckman and Opdam to the development of their theory of hypergeometric functions in higher dimension. One of the main technical problems in this context is the description of the generalized radial parts of invariant differential operators. Apart from an explicit formula for the generalized radial part of the Laplacian—an expression which was in fact the starting point for Heckman and Opdam—the other operators remain somewhat intangible.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/bdc2a06235958af74eb03061e5c3c5ce2f2b262b
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Citations in the corpus
- Positive $$L^p$$Lp-Bounded Dunkl-Type Generalized Translation Operator and Its Applications (2017, background): “By the Paley–Wiener theorem for tempered distributions (see [25, 53]) and Theorem 5.”
- REAL PALEY―WIENER THEOREMS AND LOCAL SPECTRAL RADIUS FORMULAS (2008, background): “ to be still very young and needs further testing. A context for a possible application is the following: In higher dimension, the results for complex Paley–Wiener theorems for the Dunkl transform in [43] and [60] seem to be not yet optimal, as the version for supports in compact convex invariant se”
- Explicit formulas for the Dunkl dihedral kernel and the $(\kappa, a)$-generalized Fourier kernel (2016, background): “It is known that the Dunkl kernel for positive integer κ can in principle be expressed as elementary functions, see [24] and [8].”