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


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


Zbl 0498.30010
Full Text: EuDML