## Conformal mappings and the approximation of analytic functions in domains with a quasi-conformal boundary. (Konforme Abbildungen und die Approximation analytischer Funktionen in Gebieten mit quasikonformem Rand.)(Russian)Zbl 0358.30005

This is a comprehensive paper concerning the approximation of analytic functions by algebraic polynomials in a domain of the complex plane with quasi-conformal boundary. The results given here are in a strong connection with other earlier author’s results. The main result of the paper is the following: Let $$G$$ be a bounded domain of the complex plane with quasi-conformal boundary $$L$$. One denotes by $$A(\bar{G})$$ the class of functions holomorphic in $$G$$ and continuous on $$\bar{G}$$ and by $$A^{P}(\bar{G})$$ $$(p\in \mathbb{N})$$ the class of functions $$f$$ such that $$f^{(p)}\in A(\bar{G})$$.
(Theorem 3) If $$f\in A(\bar{G})$$ then for each $$n\in \mathbb{N}$$ there exists an algebraic polynomial $$P_{n}$$ of degree $$\leq n$$, for which
$|f(z)-P_{n}(z)|\leq M \omega [\rho_{1+1/n}(z)], \quad \forall z\in \partial G$
where $$M$$ is a constant independent on $$n$$ and $$z$$, $$\omega (t) =\omega (f;t)$$ is the modulus of continuity of $$f$$ on $$\bar{G}$$, $$\rho_{1+1/n}(t)$$ is the distance from $$z\in L$$ to the level line $$L_{1+1/n}=\{z: |\Phi (z)|=1+1/n\}$$, $$w= \Phi(z)$$ being a conformal mapping of $$G$$ onto $$\{ w: |w| > 1 \}$$, $$\Phi (\infty ) = \infty$$, $$\lim_{z \to \infty} \frac{1}{z} \Phi (z) >0$$. As a consequence of this theorem combined with approximation theory inverse theorems, the author gives the following constructive characterization of the Hölder’s class $$H^{\alpha}$$ $$(\alpha \in (0,1))$$ of functions $$f$$ holomorphic in $$G$$, satisfying Hölder’s condition of order $$\alpha$$ on $$G$$ (Theorem 4): Let $$f$$ be holomorphic in $$G$$ and continuous on $$\bar{G}$$. Then $$f \in H^{\alpha}$$ $$(0 < \alpha <1)$$ if and only if for each $$n \in \mathbb{N}$$ there exists an algebraic polynomial $$P_{n}$$, of degree $$\leq n$$ satisfying the inequality: $$|f(z) - P_{n}(z)| \leq M[\rho_{1+1/n}(z)]^{\alpha}$$ . At the end of the paper an interesting result about simultaneous approximation of a function $$f$$ and its derivatives by polynomials is also derived.
Reviewer: I. Maruşciac

### MSC:

 30C35 General theory of conformal mappings 30C62 Quasiconformal mappings in the complex plane 30E10 Approximation in the complex plane
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