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

This paper discusses the approximation by de la Vall'{e}e Poussin means $V_{n}f$ 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 $f$.