The strong converse inequality for de la Vall\'{e}e Poussin means on the sphere

TL;DR

This paper establishes a strong converse inequality for de la Vallée Poussin means on the sphere, linking approximation error to the function's smoothness modulus with precise bounds.

Contribution

It proves a strong converse inequality for de la Vallée Poussin means on the sphere, providing bounds that connect approximation error with the modulus of smoothness.

Findings

Established the strong converse inequality for de la Vallée Poussin means.

Derived bounds relating approximation error to the modulus of smoothness.

Confirmed the inequality holds for functions in Lp and continuous functions on the sphere.

Abstract

This paper discusses the approximation by de la Vall\'{e}e Poussin means $V_nf$ on the unit sphere. Especially, the lower bound of approximation is studied. As a main result, the strong converse inequality for the means is established. Namely, it is proved that there are constants $C_1$ and $C_2$ such that \begin{eqnarray*} C_1\omega(f,\frac{1}{\sqrt n})_p \leq \|V_{n}f-f\|_p \leq C_2\omega(f,\frac{1}{\sqrt n})_p \end{eqnarray*} for any $p$-th Lebesgue integrable or continuous function $f$ defined on the sphere, where $\omega(f,t)_p$ is the modulus of smoothness of $f$.