Approximation by boolean sums of Jackson operators on the sphere

TL;DR

This paper establishes direct and inverse approximation theorems for Boolean sums of Jackson operators on the sphere, linking approximation quality to function smoothness and identifying the saturation order of the operators.

Contribution

It provides new theoretical results on approximation properties and saturation order of Boolean Jackson operators on the sphere, extending classical approximation theory.

Findings

Established direct and inverse theorems relating approximation error to modulus of smoothness.

Proved the saturation order of the Boolean Jackson operators is $k^{-2r}$.

Provided bounds for approximation error in terms of operator norms.

Abstract

This paper concerns the approximation by the Boolean sums of Jackson operators $\oplus^rJ_{k,s}(f)$ on the unit sphere $\mathbb S^{n-1}$ of $\mathbb{R}^{n}$. We prove the following the direct and inverse theorem for $\oplus^rJ_{k,s}(f)$: there are constants $C_1$ and $C_2$ such that \begin{equation*} C_1\|\oplus^rJ_{k,s}f-f\|_p \leq \omega^{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 $\oplus^rJ_{k,s}$ is $k^{-2r}$.