In the paper, an exact estimate of the best nonsymmetric approximation in the integral metric by the constants of continuous functions that belong to the classes $H^\omega[a,b]$ is proved. Taking into account Babenko's theorem on the connection of nonsymmetric approximation with the usual best approximation in the integral metric and the best one-sided approximations, from the proved result we obtain the exact estimate for the usual best approximation obtained by N.P. Korneichuk, and the exact estimate for the best one-sided approximation obtained by V.G. Doronin and A.A. Ligun.
We obtain upper bound of the best approximation of the classes $H^{\omega} [a, b]$ by piecewise-constant functions over uniform split in metrics of $L_{\varphi}[a, b]$ spaces, which are generated by continuous non-decreasing functions $\varphi$ that are equal to zero in zero. We study the classes of functions $\varphi$, for which the obtained bound is exact for all convex moduli of continuity.
On the nonsymmetric approximation of continuous functions in the integral metric
