Uniform polynomial approximation of analytic functions on a quasidisk
A nondecreasing function \(\mu(\cdot): \mathbb{R}_ +\to \mathbb{R}_ +\), is called a normal majorant if for some positive \(C\) and \(c\), \(\mu(t\delta)\leq Ct^ c \mu(\delta)\), for all \(t\geq 1\) and \(\delta>0\). We also assume that \(\mu(\delta)\to 0\) as \(\delta\to 0\). Let \(G\subset\mathbb{C}\) be a bounded quasidisk and let \(A(\overline{G})\) be the space of functions \(f\) which are continuous in \(\mathbb{G}\) and analytic in \(G\). The paper discusses the following questions. Describe the functions \(f\in A(\overline{G})\) for which (*) \(E_ n(f, \overline{G})=O(\mu(1/n))\), as \(n\to\infty\), where \(E_ n(f,\overline {G})\) is the degree of best uniform approximation of \(f\) by polynomials of degree not exceeding \(n\), and describe those functions \(f\in A(\overline {G})\) for which (**) \(E_ n(f, \overline{G}) \sim \mu(1/n)\), \(n=1,2,\dots\) . If \(\Omega:= \mathbb{C}\setminus G\) and \(w=\Phi(z)\) is the conformal mapping of \(\Omega\) onto \(\Delta:= \{w\): \(| w|>1\}\) normalized so that \(\Phi(\infty)=\infty\) and \(\Phi(\infty)>0\), then we extend \(\Phi\) continuously on \(\overline{\Omega}\) and define the modulus of smoothness of \(f\) on \(\partial G\) as \[ \widetilde{\omega}_ m(\delta):= \sup\{E_{m-1} (f,\Phi^{-1}(\gamma)):\;\gamma\subset \partial\Delta,\;|\gamma|\leq \delta\}, \qquad \delta>0, \] where \(|\gamma|\) is the length of the arc \(\gamma\). Then it is proved that in order that (*) would hold for \(f\in A(\overline{G})\), it is necessary that for all \(m\geq m_ 0(\mu,G)\) and it it is sufficient that for some \(m\geq 1\), we have \(\widetilde{\omega}_ m(\delta)\to 0\), as \(\delta\to 0\). Also, it is shown that in order that (**) would be satisfied, it is necessary and sufficient that \(\widetilde{\omega}_ m(\delta)\sim \mu(\delta)\), for all \(m\geq m_ 0(\mu,G)\). The results are applied to obtain a negative answer to a question of Turan about the different degrees of approximation of polynomials and rational functions \(f\in A(\overline{D})\) which cannot be analytically continued beyond \(\overline{D}\).
scientific article; zbMATH DE number 205432