This paper discusses the approximation by %semigroups of operators of class ($\mathscr{ C}_0$) on the sphere and focuses on a class of so called exponential-type multiplier operators. It is proved that such operators form a strongly continuous semigroup of contraction operators of class ($\mathscr{ C}_0$), from which the equivalence between approximation for these operators and $K$-functionals introduced by the operators is given. As examples, the constructed $r$-th Boolean of generalized spherical Abel-Poisson operator and $r$-th Boolean of generalized spherical Weierstrass operator denoted by $\oplus^r V_t^{ \gamma}$ and $\oplus^r W_t^{ \kappa}$ separately ($r$ is any positive integer, $0<\gamma,\kappa\leq1$ and $t>0$) satisfy that $\|\oplus^r V_t^{ \gamma}f - f\|_{ \mathcal{ X}}\approx \omega^{ r\gamma}(f,t^{ 1/\gamma})_{ \mathcal{ X}}$ and $\|\oplus^r W_t^{ \kappa}f - f\|_{ \mathcal{ X}}\approx \omega^{ 2r\gamma}(f,t^{ 1/(2\kappa)})_{ \mathcal{ X}}$, for all $f\in \mathcal{ X}$, where $\mathcal{ X}$ is a Banach space of continuous functions or $\mathcal{ L}^p$-integrable functions ($1\leq p<\infty$) and $\|\cdot\|_{ \mathcal{ X}}$ is the norm on $\mathcal{ X}$ and $\omega^s(f,t)_{ \mathcal{ X}}$ is the moduli of smoothness of degree $s>0$ for $f\in \mathcal{ X}$. The saturation order and saturation class of the regular exponential-type multiplier operators with positive kernels are also obtained. Moreover, it is proved that $\oplus^r V_t^{ \gamma}$ and $\oplus^r W_t^{ \kappa}$ have the same saturation class if $\gamma=2\kappa$.
Categories: math.CA