  This paper concerns the approximation by the Boolean sums of Jackson
operators $\oplus^rJ_{k,s}(f)$ on the unit sphere $\mathbb S^{n-1}$ of
$\mathbb{R}^{n}$. We prove the following the direct and inverse theorem for
$\oplus^rJ_{k,s}(f)$: there are constants $C_1$ and $C_2$ such that
\begin{equation*} C_1\|\oplus^rJ_{k,s}f-f\|_p \leq \omega^{2r}(f,k^{-1})_p \leq
C_2 \max_{v\geq k}\|\oplus^rJ_{k,s}f-f\|_p \end{equation*} for any positive
integer $k$ and any $p$th Lebesgue integrable functions $f$ defined on $\mathbb
S^{n-1}$, where $\omega^{2r}(f,t)_p$ is the modulus of smoothness of degree
$2r$ of $f$. We also prove that the saturation order for $\oplus^rJ_{k,s}$ is
$k^{-2r}$.
