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.