To the question of approximation of continuous periodic functions by trigonometric polynomials

Abstract

In the paper, it is proved that
$$1 - \frac{1}{2n} \leqslant \sup\limits_{\substack{f \in C\\f \ne const}} \frac{E_n(f)_C}{\omega_2(f; \pi/n)_C} \leqslant \inf\limits_{L_n \in Z_n(C)} \sup\limits_{\substack{f \in C\\f \ne const}} \frac{\| f - L_n(f) \|_C}{\omega_2 (f; \pi/n)_C} \leqslant 1$$
where $$$\omega_2(f; t)_C$$$ is the modulus of smoothness of the function $$$f \in C$$$, $$$E_n(f)_C$$$ is the best approximation by trigonometric polynomials of the degree not greater than $$$n-1$$$ in uniform metric, $$$Z_n(C)$$$ is the set of linear bounded operators that map $$$C$$$ to the subspace of trigonometric polynomials of degree not greater than $$$n-1$$$.