Approximation by boolean sums of Jackson operators on the sphere

Abstract

This paper concerns the approximation by the Boolean sums of Jackson operators ${\oplus }^r{J}_{k,s}(f)$ on the unit sphere ${\mathbb{S}}^{n-1}$ of ${\mathbb{R}}^n$. We prove the following the direct and inverse theorem for ${\oplus }^r{J}_{k,s}(f)$: there are constants $C_1$ and $C_2$ such that \begin{equation*} C_1\|\oplus^rJ_{k,s}f-f\|_p \leq ω^{2r}(f,k^{-1})_p \leq C_2 \max_{v\geq k}\|\oplus^rJ_{k,s}f-f\|_p \end{equation*} for any positive integer $k$ and any $p$th Lebesgue integrable functions $f$ defined on ${\mathbb{S}}^{n-1}$, where ${\omega }^{2r}(f,t{)}_p$ is the modulus of smoothness of degree $2r$ of $f$. We also prove that the saturation order for ${\oplus }^r{J}_{k,s}$ is $k^{-2r}$.