On the dependence of properties of functions on their degree of approximation by polynomials

Let $f(x)$ be a bounded $2\pi$-periodic function with modulus of continuity $\omega(\delta,f)$; let $E_n(f)$ and $H_\alpha E_n(f)$ be the minimum deviations of $f$ from the trigonometric polynomials of order $\leqslant n$, in the uniform metric and the Hausdorff metric of order $\alpha$, respectively; let
$$ \sigma_n(f,\alpha)=H_\alpha E_0(f)+\dots+H_\alpha E_{n-1}(f). $$
Then
\begin{gather*} H_\alpha E_n(f)\leqslant E_n(f)\leqslant H_\alpha E_n(f)\exp\{(3+2\sqrt2\,)\alpha\sigma_n(f,\alpha)\},\\ \omega\left(\frac1n,f\right)\leqslant\frac{\exp\{(3+2\sqrt{2})\alpha{\sigma_n}(f,\alpha)\}-1}{n\alpha}. \end{gather*}
If $H_\alpha E_n(f)\leqslant c/n\alpha$ as $n\to\infty$, then if $c<\pi$ the function $f$ is continuous almost everywhere; if $c<\pi/2$ it is continuous everywhere, and if $c<1$ we have $f\in\operatorname{Lip}\gamma(c)$, $\gamma(c)>0$.
Approximation by algebraic polynomials is also considered, and some corollaries are given.
Bibliography: 13 titles.