## 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

### MSC:

 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)

### Keywords:

Hölder class; modulus of continuity
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