Constructive descriptions of classes of functions with the help of polynomial approximations. I

Proofs of results announced earlier are given. Theorem 1, which was announced in 1976, states that a function on a domain with bounded boundary rotation can be approximated in terms of a function $\rho_1^*(z)$, which modifies the classical distance $\rho_{1/n}(z)$ for the points whose neighborhoods contain more than one arc of the level curve of the complement of the domain. Theorem 2, which was announced in 1977, provides a domain with bounded boundary rotation and a function in the analytic Hölder $\alpha$-class on the domain which cannot be approximated with precision $p_{1/n}^\alpha(z)$ by polynomials.