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.

In this note, a new proof for the positivity of Dunkl’s intertwining operator in the crystallographic case is given. It is based on an asymptotic relationship between the Opdam-Cherednik kernel and the Dunkl kernel as recently observed by M. de Jeu, and on positivity results of S. Sahi for the Heckman-Opdam polynomials and their nonsymmetric counterparts. 2000 AMS Subject Classification: 33C52,...

The goal of this paper is to present a Dunkl-Gamma type operator with the help generalization two-variable Hermite polynomials and derive its approximating properties via classical modulus continuity, second continuity Peetre’s K-functional.

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.

Dunkl operators associated with finite reflection groups generate a commutative algebra of differential-difference operators. There exists unique linear operator called intertwining which intertwines between this and the standard differential also generalization Fourier transform in context transform. In paper, we determine an integral expression for kernel, is kernel transform, all dihedral gr...