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}}^{\mathrm{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 {\oplus }^r{V}_t^{\gamma } and the rth Boolean of the generalized spherical Weierstrass operator {\oplus }^r{W}_t^k for integer r≥1 and reals γ, \kappa \in (0, 1] have errors \mathit{\parallel }{\oplus }^r{V}_t^{\gamma }f-f{\mathit{\parallel }}_X\approx {\omega }^{r\gamma }(f,t^{\mathrm{1}/\gamma }{)}_X and \mathit{\parallel }{\oplus }^r{W}_t^{k}f-f{\mathit{\parallel }}_X\approx {\omega }^{\mathrm{2}rk}{(f,t^{\mathrm{1}/(\mathrm{2}k)})}_X for all f\in X and 0≤t≤2π, where Xis the Banach space of all continuous functions or all L^pintegrable functions, 1≤p<+∞, on {\mathbb{S}}^{\mathrm{n}} with norm \mathit{\parallel }\cdot {\mathit{\parallel }}_X, and {\omega }^s{(f,t)}_Xis the modulus of smoothness of degree s>0 for f\in X. Moreover, {\oplus }^r{V}_t^{\gamma } and {\oplus }^r{W}_t^k have the same saturation class if \gamma =\mathrm{2}\kappa.

Keywords

Sphere / semigroup / approximation / modulus of smoothness / multiplier

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg