Approximation characterization of classes of functions on continua of the complex plane. (English. Russian original) Zbl 0606.30036

Math. USSR, Sb. 53, 69-87 (1986); translation from Mat. Sb., Nov. Ser. 125(167), No. 1, 70-87 (1984).
Let E be a compact connected subset of the plane \({\mathbb{C}}\) with simply connected complement \(D:={\hat {\mathbb{C}}}\setminus E\) and let \(L:=\partial D=\partial E\) their common boundary. Let \(\Phi\) map conformally D onto \(\{| w| >1\}\), \(\Phi (\infty)=\infty\), \(\Phi '(\infty)>0\). Put \[ L_ R:=\{z\in D;\quad | \Phi (z)| =R\}\quad (R>1), \]
\[ \delta_ n:=\sup \{dist(z,E);\quad z\in L_{1+1/n}\}\quad (n\geq 1). \] To illustrate results of the paper we quote Theorem 4. If any two points \(z_ 1,z_ 2\) of E can be joined by an arc \(\gamma (z_ 1,z_ 2)\) with meas \(\gamma (z_ 1,z_ 2)\leq C| z_ 1-z_ 2|\) \((C=const\geq 1)\), then \(f\in H^{\alpha}(E)\) (Hölder class) if and only if there exists a sequence of polynomials \((P_ n)_{n\geq 1}\) such that deg \(P_ n\leq n\), \[ | f(z)-P_ n(z)| \leq C_ 1\omega (\delta_ n),\quad | P_ n'(z)| \leq C_ 2\omega (\delta_ n)\delta_ n^{-1} \] for all \(z\in E\), where \(C_ 1,C_ 2\) are positive constants and \(\omega\) is the modulus of continuity of f.
Reviewer: J.Siciak


30E10 Approximation in the complex plane
41A10 Approximation by polynomials
41A27 Inverse theorems in approximation theory
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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