Approximation by Semigroups of Spherical Operators

Abstract: 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 $\gamma=2\kappa$.