For a family of weight functions, hκ, invariant under a finite reflection group on R, analysis related to the Dunkl transform is carried out for the weighted L spaces. The generalized translation operator and the weighted convolution are studied in detail in L(R, h2κ) and the result is used to study the summability of the inverse Dunkl transform, including the Poisson integrals and the Bochner-...

For a family of weight functions, hκ, invariant under a finite reflection group on R, analysis related to the Dunkl transform is carried out for the weighted L spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a m...

using a generalized spherical mean operator, we obtain the generalizationof titchmarsh's theorem for the dunkl transform for functions satisfyingthe lipschitz condition in l2(rd;wk), where wk is a weight function invariantunder the action of an associated reection groups.

The Al-Salam & Carlitz polynomials are q-generalizations of the classical Hermite polynomials. Multivariable generalizations of these polynomials are introduced via a generating function involving a multivariable hypergeometric function which is the q-analogue of the type-A Dunkl integral kernel. An eigenoperator is established for these polynomials and this is used to prove orthogonality with ...

The aim of this work is to introduce a novel concept, Riesz–Dunkl fractional derivatives, within the context Dunkl-type operators. A particularly noteworthy revelation that when specific parameter κ equals zero, derivative smoothly reduces both well-known Riesz and second-order derivative. Furthermore, we new concept: Sobolev space. This space defined characterized using versatile framework Dun...

The Al-Salam & Carlitz polynomials are q-generalizations of the classical Hermite polynomials. Multivariable generalizations of these polynomials are introduced via a generating function involving a multivariable hypergeometric function which is the q-analogue of the type-A Dunkl integral kernel. An eigenoperator is established for these polynomials and this is used to prove orthogonality with ...

For a family of weight functions, hκ, invariant under a finite reflection group on R, analysis related to the Dunkl transform is carried out for the weighted L spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a m...

We study the Sobolev spaces of exponential type associated with the Dunkl-Bessel Laplace operator. Some properties including completeness and the imbedding theorem are proved. We next introduce a class of symbols of exponential type and the associated pseudodifferential-difference operators, which naturally act on the generalized Dunkl-Sobolev spaces of exponential type. Finally, using the theo...

For a family of weight functions, hκ, invariant under a finite reflection group on R, analysis related to the Dunkl transform is carried out for the weighted L spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a m...

In this paper we prove the absolute continuity of the representing measures of the Dunkl intertwining operator and of its dual. Next we present some applications of this result. Key word : Dunkl intertwining operator and its dual. Absolute continuity of the representing measures. MSC (2000) : 33C80, 51F15, 44A15. Introduction We consider the differential-difference operators on R introduced by ...