Approximation by Semigroups of Spherical Operators

Abstract

This paper discusses the approximation by %semigroups of operators of class (${\mathcal{C}}_0$) 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 (${\mathcal{C}}_0$), 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, $0<\gamma ,\kappa \le 1$ and $t>0$) satisfy that $\Vert {\oplus }^r{V}_t^{\gamma }f-f{\Vert }_{\mathcal{X}}\approx {\omega }^{r\gamma }(f,t^{1/\gamma }{)}_{\mathcal{X}}$ and $\Vert {\oplus }^r{W}_t^{\kappa }f-f{\Vert }_{\mathcal{X}}\approx {\omega }^{2r\gamma }(f,t^{1/(2\kappa )}{)}_{\mathcal{X}}$, for all $f\in \mathcal{X}$, where $\mathcal{X}$ is a Banach space of continuous functions or ${\mathcal{L}}^p$-integrable functions ($1\le p<\infty$) and $\Vert \cdot {\Vert }_{\mathcal{X}}$ 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 $\gamma =2\kappa$.