On one extremal property of Korovkin's means

Abstract

We point out that$$\inf\limits_{L \in L_n} \sup\limits_{\substack{f \in C_{2\pi}\\f \ne const}} \frac{\max \| f(x) - L(f, x) \|}{\omega^*_2(f, \pi/n + 1)} = \frac{1}{2}$$where $C_{2\pi}$ is the space of periodic continuous functions on real domain, $L_n$ is the set of linear operators that map $C_{2\pi}$ to the set of trigonometric polynomials of order no greater than $n$ ($n = 0,1,\ldots$), $\omega_2(f, t) = \sup\limits_{x, |h| \leqslant t} |f(x-h) - 2f(x) + f(x+h)|$, $\omega^*_2(f, t)$ is the concave hull of the function $\omega_2(f, t)$. In this equality, the infimum is attained for Korovkin's means.