The aim of this paper is to prove new quantitative uncertainty principle for the generalized Fourier transform connected with a Dunkl type operator on the real line. More precisely we prove An Lp-Lq-version of Morgan's theorem.

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.

In this paper, using a generalized Jacobi-Dunkl translation operator, we prove a generalization of Titchmarsh’s theorem for functions in the k-JacobiDunkl-Lipschitz class defined by the finite differences of order k ∈ N∗ and Sobolev spaces associated with the Jacobi-Dunkl operator.

It is an open conjecture that generalized Bessel functions associated with root systems have a positive product formula for non-negative multiplicity parameters of the associated Dunkl operators. In this paper, a partial result towards this conjecture is proven, namely a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel. Radia...

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. Furtherm...

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...

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...