Inversion of the Dunkl-Hermite Semigroup

Néjib Ben Salem ; Walid Nefzi

Bulletin of the Malaysian Mathematical Society, Tome 35 (2012) no. 2

Résumé

Let $\{e^{-c\mathcal{H}^\alpha}/\Re c\geq 0\}$ be the Dunkl-Hermite semigroup on the real line $\mathbb{R}$, defined by

$[e^{-c\mathcal{H}^\alpha}f](x)=\int_\mathbb{R}\mathcal{K}_c^\alpha(x,\xi)f(\xi)d\mu_\alpha(\xi)\;, \quad x\in\mathbb{R}\;,$

where $\mathcal{K}_c^\alpha(x,\xi)=\sum_{n=0}^\infty e^{-cn} H_n^\alpha(x)H_n^\alpha(\xi)$. Here, $H_n^\alpha, n\in \mathbb{N}$, are the Dunkl-Hermite polynomials which are the eigenfunctions of the operator $D_\alpha^2-2x{d}/{dx}$, $D_\alpha$ being the Dunkl operator on the real line. For $\Re c>0$, we give a representation for inverting the semigroup. Next, we extend $e^{-c\mathcal{H}^\alpha}$ and we give an integral representation of it for $\Re c0$. Moreover, in this last case, we characterize the domain in which $e^{-c\mathcal{H}^\alpha}$ is well defined.

Détail

Classification : 47D03, 47B38, 46E20, 33C45.