## Geometric structure of domains and direct theorems of constructive theory of functions.(Russian)Zbl 0578.30030

Let G be a bounded domain on the complex plane $${\mathbb{C}}$$ whose boundary $$L:=\partial G$$ is a Jordan curve. Let g be the conformal mapping of $${\hat {\mathbb{C}}}\setminus \bar G$$ onto $$\{| w| >1\}$$ with $$g(\infty)=\infty$$, $$g'(\infty)>0$$. If $$R>1$$ we put $$L_ R:=\{| g(z)| =R\}$$, $$\rho_ R(z):=dist(z,L_ R)$$. Let $$W^ rCH^{\omega}$$ denote the set of all functions f holomorphic in G such that $$f^{(s)}$$, $$s=0,...,r$$, are uniformly continuous in G and $| f^{(r)}(z)-f^{(r)}(\zeta)| \leq C\omega (| z-\zeta |),z,\zeta \in G,$ where $$C=const>0$$, and $$\omega$$ is a fixed positive increasing function with $$\omega (+0)=0$$, $$\omega (t\delta)\leq C_ 1t\omega (\delta)$$ for $$\delta >0$$, $$t>1$$ $$(C_ 1=const>0)$$. The author says that G satisfies the D-property, if for every function $$f\in W^ rCH^{\omega}$$ there exists a sequence of polynomials $$(P_ n)_{n\geq r+1}$$ such that deg $$P_ n\leq n$$ and $| f(z)-P_ n(z)| \leq C_ 2\rho^ r_{1+1/n}(z)\omega (\rho_{1+1/n}(z)),\quad z\in L;\quad C_ 2=const>0.$ It is shown that the domains G of the author’s class $$H^*$$ (introduced in his earlier paper [Ukr. Mat. Zh. 33, 723-727 (1981; Zbl 0498.30010)] have the D-property. Moreover the D-property of G is very close to the fact that G is a member of $$H^*$$.
Reviewer: J.Siciak

### MSC:

 30E10 Approximation in the complex plane 41A10 Approximation by polynomials 30C62 Quasiconformal mappings in the complex plane

### Keywords:

level lines of conformal mapping

Zbl 0498.30010
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