Approximation by semigroup of spherical operators

Abstract

This paper concerns about the approximation by a class of positive exponential type multiplier operators on the unit sphere $\mathbb{S}^n$ of the (n + 1)-dimensional Euclidean space for n ⩾ 2. We prove that such operators form a strongly continuous contraction semigroup of class $(C_0 )$ and show the equivalence between the approximation errors of these operators and the K-functionals. We also give the saturation order and the saturation class of these operators. As examples, the rth Boolean of the generalized spherical Abel-Poisson operator ⊕^r V ^γ _t and the rth Boolean of the generalized spherical Weierstrass operator ⊕^r W ^κ _t for integer r ⩾ 1 and reals γ, κ ∈ (0, 1] have errors $\left\| { \oplus ^r V_t^\gamma f - f} \right\|_X \asymp \omega ^{r\gamma } (f,t^{1/\gamma } )_X$ and $\left\| { \oplus ^r W_t^\kappa f - f} \right\|_X \asymp \omega ^{r\kappa } (f,t^{1/(2\kappa )} )_X$ for all f ∈ $X$ and 0 ⩽ t ⩽ 2π, where $X$ is the Banach space of all continuous functions or all ℒ ^p integrable functions, 1 ⩽ p < +∞, on $\mathbb{S}^n$ with norm $\left\| \cdot \right\|_X$, and $\omega ^s (f,t)_X$ is the modulus of smoothness of degree s > 0 for f ∈ $X$. Moreover, ⊕^r V ^γ _t and ⊕^r W ^κ _t have the same saturation class if γ = 2κ.