The Direct and Converse Inequalities for Jackson-Type Operators on Spherical Cap

TL;DR

This paper constructs Jackson-type operators for approximation on spherical caps, establishing direct and inverse inequalities, saturation theorems, and analyzing their approximation order, which advances understanding of approximation theory on spherical domains.

Contribution

The paper introduces new Jackson-type operators tailored for spherical caps and proves their approximation properties, including inequalities and saturation order, extending classical results from the full sphere to caps.

Findings

Established direct and inverse inequalities for the operators.

Proved the saturation order of approximation is O(k^{-2}).

Demonstrated the effectiveness of the operators in approximating functions on spherical caps.

Abstract

Approximation on the spherical cap is different from that on the sphere which requires us to construct new operators. This paper discusses the approximation on the spherical cap. That is, so called Jackson-type operator $\{J_{k,s}^m\}_{k=1}^{\infty}$ is constructed to approximate the function defined on the spherical cap $D(x_0,\gamma)$. We thus establish the direct and inverse inequalities and obtain saturation theorems for $\{J_{k,s}^m\}_{k=1}^{\infty}$ on the cap $D(x_0,\gamma)$. Using methods of $K$-functional and multiplier, we obtain the inequality \begin{eqnarray*} C_1\:\| J_{k,s}^m(f)-f\|_{D,p}\leq \omega^2\left(f,\:k^{-1}\right)_{D,p} \leq C_2 \max_{v\geq k}\| J_{v,s}^m(f) - f\|_{D,p} \end{eqnarray*} and that the saturation order of these operators is $O(k^{-2})$, where $\omega^2\left(f,\:t\right)_{D,p}$ is the modulus of smoothness of degree 2, the constants $C_1$ and $C_2$ are independent of $k$ and $f$.