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Approximation by Semigroups of Spherical Operators

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 $\oplus^r V_t^γ$ and $\oplus^r W_t^κ$ separately ($r$ is any positive integer, $0<γ,κ\leq1$ and $t>0$) satisfy that $\|\oplus^r V_t^γf - f\|_{\mathcal{X}}\approx ω^{rγ}(f,t^{1/γ})_{\mathcal{X}}$ and $\|\oplus^r W_t^κf - f\|_{\mathcal{X}}\approx ω^{2rγ}(f,t^{1/(2κ)})_{\mathcal{X}}$, 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 $ω^s(f,t)_{\mathcal{X}}$ 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 $\oplus^r V_t^γ$ and $\oplus^r W_t^κ$ have the same saturation class if $γ=2κ$.