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

Belyĭ, V. I.

Mat. Sb., N. Ser. 102(144), 331-361 (1977).

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