Multiple modules of continuity and the best approximations of periodic functions in metric spaces

Abstract

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 $$\sup_n \sup_{f\in L_{\Psi}, f\not = \text{const}} \frac{E_{n-1}(f)_{\Psi} }{\omega_k \Bigl(f, \frac{\pi}n \Bigr)_{\Psi}} < \infty,$$ 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: $E_{n-1}(f)_{\Psi} \asymp \omega_k \Bigl(f, \frac{\pi}n \Bigr)_{\Psi}.$