Convergence in Capacity of Rational Approximants of Meromorphic Functions

Hans-Peter Blatt

Abstract: Let $f$ be meromorphic on the compact set $E\subset\mathbb{C}$ with maximal Green domain of meromorphy $E_{\rho(f)}$, $\rho(f)<\infty$. We investigate rational approximants with numerator degree $\leq n$ and denominator degree $\leq m_n$ for $f$. We show that the geometric convergence rate on $E$ implies convergence in capacity outside $E$ if $m_n=o(n)$ as $n\to\infty$. Further, we show that the condition is sharp and that the convergence in capacity is uniform for a subsequence $\Lambda\subset
mathbb{N}$.

Keywords: rational approximation; convergence in capacity

Classification (MSC2000): 41A20; 41A24; 30E10

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