Dunkl harmonic analysis and fundamental sets of functions on the unit sphere

Abstract

Using Dunkl theory, we introduce into consideration some weighted $L_p$-spaces on $[-1,1]$ and on the unit Euclidean sphere ${\mathbb{S}}^{d-1}$, $d\ge 2$. Then we define a family of linear bounded operators $\{V_{\kappa }^p(x):x\in {\mathbb{S}}^{d-1}\}$ acting from the $L_p$-space on $[-1,1]$ to the $L_p$-space on ${\mathbb{S}}^{d-1}$, $1\le p<\infty$. We establish a necessary and sufficient condition for a function $g$ belonging to the $L_p$-space on $[-1,1]$ such that the family of functions $\{V_{\kappa }^p(x;g):x\in {\mathbb{S}}^{d-1}\}$ is fundamental in the $L_p$-space on ${\mathbb{S}}^{d-1}$.