Paley-Wiener theorems for the Dunkl transform

Abstract

We conjecture a geometrical form of the Paley-Wiener theorem for the Dunkl transform and prove three instances thereof, one of which involves a limit transition from Opdam's results for the graded Hecke algebra. 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, which implies that the generalized Bessel functions coincide with the spherical functions. In this context, the restriction of Dunkl's intertwining operator to the invariants can be interpreted in terms of the Abel transform. Using shift operators we also show that, for certain values of the multiplicities of the restricted roots, the Abel transform is essentially inverted by a differential operator.

LaTeX, 26 pages, no figures. References updated and minor changes, mathematically identical to the first version. To appear in Trans. Amer. Math. Soc

References in corpus (5)

• Asymptotic analysis for the Dunkl kernel
• Subspaces with equal closure
• Singular Polynomials and Modules for the Symmetric Groups
• Singular polynomials for the symmetric groups
• Convolution operator and maximal function for Dunkl transform

Cited by in corpus (20)

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• Elementary proofs of Paley-Wiener theorems for the Dunkl transform on the real line
• A limit transition from the Heckman-Opdam hypergeometric functions to the Whittaker functions associated with root systems
• A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian
• Convolution operator and maximal function for Dunkl transform
• Positive $L^p$-bounded Dunkl-type generalized translation operator and its applications
• Uncertainty principles for eventually constant sign bandlimited functions
• Sonine formulas and intertwining operators in Dunkl theory
• Radial components, prehomogeneous vector spaces, and rational Cherednik algebras
• Positivity of Dunkl's intertwining operator via the trigonometric setting
• Inversion Formulas for the Dunkl Intertwining Operator and Its Dual on Spaces of Functions and Distributions
• A formula for the hypergeometric function of type $B{C}_n$
• Hardy--Littlewood inequalities for the Heckman--Opdam transform
• Sharp Estimates of Radial Dunkl and Heat Kernels in the Complex Case $A_n$
• Wave functions for quantum integrable particle systems via partial confluences of multivariate hypergeometric functions
• Fractional smoothness in $L^p$ with Dunkl weight and its applications
• The Dunkl kernel and intertwining operator for dihedral groups
• Three results in Dunkl theory
• Dunkl intertwining operator for symmetric groups
• Multiple-particle diffusion processes from the viewpoint of Dunkl operators: relaxation to the steady state