Frontiers of Mathematics in China

, Volume 9, Issue 2, pp 387–416 | Cite as

Approximation by semigroup of spherical operators

Research Article
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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 ⊕rVtγ and the rth Boolean of the generalized spherical Weierstrass operator ⊕rWtκ 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 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, ⊕rVtγ and ⊕rWtκ have the same saturation class if γ = 2κ.

Keywords

Sphere semigroup approximation modulus of smoothness multiplier 

MSC

42C10 41A25 

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Copyright information

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsChina Jiliang UniversityHangzhouChina

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