Geodesic
Convergence in Capacity of Rational Approximants of Meromorphic Functions
Publications de l'Institut Mathématique, _N_S_96 (2014) no. 110, p. 31

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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 $\Lambda\subset\\mathbb{N}$.
Classification : 41A20 41A24 30E10
Keywords: rational approximation, convergence in capacity 
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     author = {Hans-Peter Blatt},
     title = {Convergence in {Capacity} of {Rational} {Approximants} of {Meromorphic} {Functions}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {31 },
     publisher = {mathdoc},
     volume = {_N_S_96},
     number = {110},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_2014_N_S_96_110_a3/}
}
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Hans-Peter Blatt. Convergence in Capacity of Rational Approximants of Meromorphic Functions. Publications de l'Institut Mathématique, _N_S_96 (2014) no. 110, p. 31 . http://geodesic.mathdoc.fr/item/PIM_2014_N_S_96_110_a3/