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, ‎using a generalized Dunkl translation operator‎, ‎we obtain a generalization of Titchmarsh's Theorem for the Dunkl transform for functions satisfying the$(psi,p)$-Lipschitz Dunkl condition in the space $mathrm{L}_{p,alpha}=mathrm{L}^{p}(mathbb{R},|x|^{2alpha+1}dx)$‎, ‎where $alpha>-frac{1}{2}$.  

These lecture notes are intended as an introduction to the theory of rational Dunkl operators and the associated special functions, with an emphasis on positivity and asymptotics. We start with an outline of the general concepts: Dunkl operators, the intertwining operator, the Dunkl kernel and the Dunkl transform. We point out the connection with integrable particle systems of Calogero-MoserSut...

In this paper, we consider a q-analogue of the Dunkl operator on R, we define and study its associated Fourier transform which is a q-analogue of the Dunkl transform. In addition to several properties, we establish an inversion formula and prove a Plancherel theorem for this q-Dunkl transform. Next, we study the q-Dunkl intertwining operator and its dual via the q-analogues of the Riemann-Liouv...

The operator that intertwines between the $\mathbb{Z}_2$ - Dunkl and derivative is shown to have a realization in terms of oscillator operators one dimension. This observation rests on fact intertwining maps Hermite polynomials generalized polynomials.

For a finite reflection group on R , the associated Dunkl operators are parametrized firstorder differential-difference operators which generalize the usual partial derivatives. They generate a commutative algebra which is – under weak assumptions – intertwined with the algebra of partial differential operators by a unique linear and homogeneous isomorphism on polynomials. In this paper it is s...

In this paper, several direct and inverse theorems in terms of the best approximations functions moduli smoothness are proved concerning approximation from space $$\mathbb {L}_{2}^{(\alpha ,\beta )}$$ by partial sums Jacobi-Dunkl series. For purpose, we use generalized translation operator which was defined Vinogradov.

In 1961, Bargmann introduced the classical Fock space $$\mathscr {F}(\mathbb {C}^{d})$$ and in 1984, Cholewinsky generalized {F}_{\alpha ,e}(\mathbb . These two spaces are aim of many works, have applications mathematics, physics, quantum mechanics. this work, we introduce study }(\mathbb associated to Dunkl operators $$T_{\alpha _{j}}$$ with $$\alpha _{j}>-1/2$$ for all $$j=1,\ldots ,d$$ This ...

The classical theory of Hardy spaces on n has received an important impetus from the work of Fefferman and Stein, Lu and Yang 1, 2 . Their work resulted in many applications involving sharp estimates for convolution and multiplier operators. By using the technique of Herz-type Hardy spaces for the Dunkl operator Λα, we are attempting in this paper to study the Dunkl convolution operators, andwe...