Dunkl Theory Research Articles

Hypoelliptic convolution equations in $\cal{S}' (\Bbb R)$ for the Dunkl theory on $\Bbb R$

Hypoelliptic convolution equations in $\cal{S}' (\Bbb R)$ for the Dunkl theory on $\Bbb R$

Segal-Bargmann transforms associated with finite Coxeter groups

Using a polarization of a suitable restriction map, and heat-kernel analysis, we construct a generalized Segal-Bargmann transform associated with every finite Coxeter group G on ℝN. We find the integral representation of this transform, and we prove its unitarity. To define the Segal-Bargmann transform, we introduce a Hilbert space Open image in new window of holomorphic functions on Open image in new window with reproducing kernel equal to the Dunkl-kernel. The definition and properties of Open image in new window extend naturally those of the well-known classical Fock space. The generalized Segal-Bargmann transform allows to exhibit some relationships between the Dunkl theory in the Schrodinger model and in the Fock model. Further, we prove a branching decomposition of Open image in new window as a unitary Open image in new window-module and a general version of Hecke's formula for the Dunkl transform.

GENERALIZED FOURIER EXPANSIONS OF DIFFERENTIABLE FUNCTIONS ON THE SPHERE

We show that the Fourier expansion in spherical $h$ -harmonics (from Dunkl's theory) of a function $f$ on the sphere converges uniformly to $f$ if this function is sufficiently differentiable.

Intertwining Operator and h-Harmonics Associated With Reflection Groups

AbstractWe study the intertwining operator and h-harmonics in Dunkl's theory on h–harmonics associated with reflection groups. Based on a biorthogonality between the ordinary harmonics and the action of the intertwining operator V on the harmonics, the main result provides a method to compute the action of the intertwining operator V on polynomials and to construct an orthonormal basis for the space of h-harmonics.