Opdam functions: product formula and convolution structure in dimension 1
Résumé
$G_\lambda^{(\alpha,\beta)}(x)$ be the eigenfunctions of the Dunkl-Cherednik operator $T^{(\alpha,\beta)}$ on $\mathbb{R}$, with $\alpha\geq \beta\geq -{1\over 2}.$ In this paper we express the product $G_\lambda^{(\alpha,\beta)}(x)G_\lambda^{(\alpha,\beta)}(y)$ as an integral in terms of $G_\lambda^{(\alpha,\beta)}(z)$ with an explicit kernel. In general this kernel is not positive. Furthermore, by taking the so-called rational limit, we recover the product formula for the Dunkl kernels proved in \cite{Ros}. We then define and study a convolution structure associated to $G_\lambda^{(\alpha,\beta)}.$
Origine : Fichiers produits par l'(les) auteur(s)