Maximally convergent rational approximants of meromorphic functions
Volume 107 / 2015
Abstract
Let $f$ be meromorphic on the compact set $E \subset \CC$ with maximal Green domain of meromorphy $E_{\rho(f)}$, $\rho(f) \lt \infty$. We investigate rational approximants $r_{n,m_n}$ of $f$ on $E$ with numerator degree $\leq n$ and denominator degree $\leq m_n$. We show that a geometric convergence rate of order $\rho(f)^{-n}$ on $E$ implies uniform maximal convergence in $m_1$-measure inside $E_{\rho(f)}$ if $m_n = o(n/\log n)$ as $n \rightarrow \infty$. If $m_n = o(n)$, $n \rightarrow \infty$, then maximal convergence in capacity inside $E_{\rho(f)}$ can be proved at least for a subsequence $\Lambda \subset \mathbb N$. Moreover, an analogue of Walsh’s estimate for the growth of polynomial approximants is proved for $r_{n,m_n}$ outside $E_{\rho(f)}$.
