Rates of best uniform rational approximation of analytic functions by ray sequences of rational functions (Q1288971)

Let \(E \) be a compact subset of the complex plane \({\mathbb C}\) and define \(\rho_{n,m}\) as the distance of a function \(f\), holomorphic on \(E\), in the uniform metric on \(E\) from the class \(R_{n,m}\) of rational functions of order \((n,m)\). The authors consider the behaviour of \(\rho_{n,m(n)}\) for ray sequences, where \(m(n)/n\to \theta \in (0,1]\) as \(n \to \infty\). The main results: If \(f\) is holomorphic on \(\overline{{\mathbb C}} \setminus F\), where \(F\) is a compact set in the extended plane \(\overline{{\mathbb C}}\) such that \(E\cap F = \emptyset\), then \(\limsup \rho_{n,m(n)} ^{1/(n+m(n))} \leq \rho ^{-1/(1+\theta)}\) and \(\liminf \rho_{n,m(n)} ^{1/(n+m(n))} \leq \rho ^{-2/(2-\theta)(1+\theta)}\) as \(n\to \infty\). The number \(\rho\) on the right side can be calculated from a minimal energy problem. For \(\theta =1\), this constant is \(\rho =\exp(1/C(E,F))\) with the capacity \(C(E,F)\) of the condenser formed by the sets \(E\) and \(F\), and the estimates reduced to known results of Walsh, Gonchar and the first author.