@article{Pichugov_2018, title={Multiple modules of continuity and the best approximations of periodic functions in metric spaces}, volume={70}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/1588}, abstractNote={It is proved that, under the condition $M_{\Psi} \Bigl( \frac 12\Bigr) < 1$, where $M_{\Psi}$ is a stretching function $\Psi$ in the space $L_{\Psi}$ , the Jackson inequalities $$\sup_n \sup_{f\in L_{\Psi}, f ot = \text{const } \frac{E_{n-1}(f)_{\Psi} }{\omega_k \Bigl(f, \frac{\pi}n \Bigr)_{\Psi } < \infty,$$ are true; here, $E_{n-1}(f)_{\Psi}$ is the best approximation of $f$ by trigonometric polynomials of degree at most $n - 1$ and $\omega_k \Bigl(f, \frac{\pi}n \Bigr)_{\Psi}$ is the modulus of continuity of $f$ of order $k$, $k \in N$. We study necessary and sufficient conditions for the function $f$ under which the following relation is true: $E_{n-1}(f)_{\Psi} \asymp \omega_k \Bigl(f, \frac{\pi}n \Bigr)_{\Psi}.$}, number={5}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Pichugov, S. A.}, year={2018}, month={May}, pages={699-707} }