Convergence in Capacity of Rational Approximants of Meromorphic Functions

Let f be meromorphic on the compact set E ⊂ C with maximal Green domain of meromorphy Eρ(f), ρ(f) < ∞. We investigate rational approximants with numerator degree 6 n and denominator degree 6 mn for f . We show that the geometric convergence rate on E implies convergence in capacity outside E if mn = o(n) as n → ∞. Further, we show that the condition is sharp and that the convergence in capacity is uniform for a subsequence Λ ⊂ N.