The impact of poles on the convergence of best rational approximants

Kurzfassung/Abstract

The convergence behaviour of best uniform rational approximants $r^*_{n,m}$ with numerator degree $n$ and denominator degree $m$ on a compact set $E$ is investigated for functions $f$ continuous on $E$ if ray sequences above the diagonal of the Walsh table are considered, i. e. sequences $\{n,m(n)\}^\infty_{n=1}$ with
$$
\underset{n \rightarrow \infty}{\lim \inf} \, \frac{n}{m(n)}
\geq c > 1.
$$
If the sequence $\{r^*_{n,m(n)}\}^\infty_{n=1}$ satisfies a pole condition $(P_\tau)$, $\tau > 1$, i. e. for any $(n,m(n))$ there exists a pole $\xi_n$ of $r^*_{n,m(n)}$ with $G(\xi_n,\infty) > \log \tau$ where $G(z,\infty)$ is Green´s function of $\overline{\CC} \setminus E$ with pole at $\infty$, then only two situations can occur:
\begin{enumerate}[(1)] \item For any continuum $S \subset \CC \setminus E$
$$
\underset{n \rightarrow \infty}{\lim \sup} \, \|r^*_{n,m(n)}\|_S =
\infty,
$$
\end{enumerate}
or
\begin{enumerate}[(2)]
\item $\underset{n \rightarrow \infty}{\lim \sup} \, \|f -
r^*_{n,m(n)}\|^{1/n}_E < 1$.
\end{enumerate}