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 ...

in this paper, using a generalized dunkl translation operator, we obtain an analog of titchmarsh's theorem for the dunkl transform for functions satisfying the lipschitz-dunkl condition in $mathrm{l}_{2,alpha}=mathrm{l}_{alpha}^{2}(mathbb{r})=mathrm{l}^{2}(mathbb{r}, |x|^{2alpha+1}dx), alpha>frac{-1}{2}$.

In this article, we establish first a geometric Paley–Wiener theorem for the Dunkl transform in the crystallographic case. Next we obtain an optimal bound for the L p → L norm of Dunkl translations in dimension 1. Finally we describe more precisely the support of the distribution associated to Dunkl translations in higher dimension.

We study some quadratic algebras which are appeared in the low–dimensional topology and Schubert calculus. We introduce the Jucys–Murphy elements in the braid algebra and in the pure braid group, as well as the Dunkl elements in the extended affine braid group. Relationships between the Dunkl elements, Dunkl operators and Jucys–Murphy elements are described.

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

In this paper we define and study the Dunkl convolution product and the Dunkl transform on spaces of distributions on R. By using the main results obtained, we study the hypoelliptic Dunkl convolution equations in the space of distributions.