*(English)*Zbl 1073.30029

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 ${\parallel \xb7-\xb7\parallel}_{E}$ on $E$ by the set ${R}_{n}$ of rational functions of order at most $n$, that is, ${\rho}_{n}(f,E)={inf}_{r\in {R}_{n}}{\parallel f-r\parallel}_{E}$. 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}\ne \varnothing $, $\overline{C}\setminus K$ is connected and $f$ is analytic in ${\Omega}$ and nonrational, then

in particular, ${lim\; inf}_{n\to \infty}{({\rho}_{n}(f,K)/{\rho}_{n}(f,E))}^{1/n}\le exp(-2/C(\partial E,K))$. Here $C(\partial E,K)$ denotes the condenser capacity associated with the condenser $(\partial E,K)$. The main result is derived by using the Hankel operator ${A}_{f,G\left(\epsilon \right)}:\phi \in {E}_{2}\left(\partial G\left(\epsilon \right)\right)\mapsto P\left(\phi f\right)\in {L}_{2}\left(\partial G\left(\epsilon \right)\right)\ominus {E}_{2}\left(\partial G\left(\epsilon \right)\right)$ associated to $f$ and to an adequate family of domains $\left\{G\right(\epsilon ):\phantom{\rule{0.166667em}{0ex}}0<\epsilon <1\}$, where, for every domain $G$ of $C$, ${E}_{2}\left(G\right)$ denotes its Smirnov class and $P$ is the orthogonal projection from ${L}_{2}\left(\partial G\right)$ onto ${E}_{2}\left(G\right)$.

##### MSC:

30E10 | Approximation in the complex domain |

41A20 | Approximation by rational functions |

41A50 | Best approximation, Chebyshev systems |