  Using Dunkl theory, we introduce into consideration some weighted
$L_p$-spaces on $[-1,1]$ and on the unit Euclidean sphere $\mathbb{S}^{d-1}$,
$d\geq 2$. Then we define a family of linear bounded operators
$\{V_\kappa^p(x)\colon x\in\mathbb{S}^{d-1}\}$ acting from the $L_p$-space on
$[-1,1]$ to the $L_p$-space on $\mathbb{S}^{d-1}$, $1\leq p<\infty$. 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
$\{V_\kappa^p(x;g)\colon x\in\mathbb{S}^{d-1}\}$ is fundamental in the
$L_p$-space on $\mathbb{S}^{d-1}$.
