CONVERGENCE IN CAPACITY OF RATIONAL APPROXIMANTS OF MEROMORPHIC FUNCTIONS

Let f be meromorphic on the compact set E subset of C with maximal Green domain of meromorphy E rho(f), rho(f) < infinity. We investigate rational approximants with numerator degree <= n and denominator degree <= 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 -> infinity. Further, we show that the condition is sharp and that the convergence in capacity is uniform for a subsequence Lambda subset of N.