On best rational approximation of analytic functions
The main result contained in this paper is described as follows. Let \(E\) be a compact set in the extended complex plane \(\overline{C}\), \(n\) be a nonnegative integer and \(f:E \to C\) be continuous. Let us consider the best rational approximation \(\rho_n(f,E)\) of \(f\) in the uniform metric \(\| \cdot - \cdot \| _E\) on \(E\) by the set \(R_n\) of rational functions of order at most \(n\), that is, \(\rho_n(f,E) = \inf_{r \in R_n} \| f-r\| _E\). Denote by \(\Omega\) the interior of \(E\) and assume that \(K \subset \overline{C}\) is a compact subset with \(K \subset \Omega\). If \(E\) has nonempty connected complement, \(\Omega \neq \emptyset\), \(\overline{C} \setminus K\) is connected and \(f\) is analytic in \(\Omega\) and nonrational, then \[ \limsup_{n \to \infty} \left( \prod_{k=0}^n \rho_k(f,K)/ \prod_{k=0}^n \rho_k(f,E) \right)^{1/n^2} \leq \exp (-1/C(\partial E,K)); \] in particular, \(\liminf_{n \to \infty} (\rho_n (f,K)/ \rho_n (f,E))^{1/n} \leq \exp (-2/C(\partial E,K))\). He\-re \(C(\partial E,K)\) denotes the condenser capacity associated with the condenser \((\partial E,K)\). The main result is derived by using the Hankel operator \(A_{f,G(\varepsilon )}: \varphi \in E_2(\partial G(\varepsilon )) \mapsto P(\varphi f) \in L_2(\partial G(\varepsilon )) \ominus E_2(\partial G(\varepsilon ))\) associated to \(f\) and to an adequate family of domains \(\{G(\varepsilon ): \, 0 < \varepsilon < 1\}\), where, for every domain \(G\) of \(C\), \(E_2(G)\) denotes its Smirnov class and \(P\) is the orthogonal projection from \(L_2(\partial G)\) onto \(E_2(G)\).
scientific article; zbMATH DE number 2160169