On the uniform polynomial approximation in complex domain summary

The class of the so-called Faber domaines is introduced.This class contains all convex domains, all domains with the piecewise smooth boundary with the finite number of angular points and all domains with the boundary of bounded rotation. The article yields a complete description of the analytic functions $f$ in the Faber domain $G$ with
$$ E_n(f;G)=O(n^{-s}),\quad s>0,\quad n\to\infty, $$
where $E_n(f;G)$ is the best polynomial approximation of degree $\leq n$. In the case of the piecewise smooth boundary with angular points this description admits a reformulation in standard metric terms.