Overconvergence of Rational Approximants of Meromorphic Functions

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Let E be a compact set in \(\mathbb{C}\) with regular connected complement Ω, and let f be meromorphic on E with maximal Green domain of meromorphy Eρ(f), ρ(f) < ∞. We investigate rational approximants \(r_{n,m_{n}}\) of f with numerator degree ≤ n and denominator degree ≤ m n and deduce overconvergence properties from geometric convergence rates of \(f - r_{n,m_{n}}\) near the boundary of E if n → ∞ and m n = o(n) (resp. m n = o(n∕logn)) as n → ∞. Moreover, results about the limiting distribution of the zeros of \(r_{n,m_{n}}\), as well as for the distribution of the interpolation points of multipoint Pade approximation can be derived. Hereby, well-known results for polynomial approximation of holomorphic functions are generalized for rational approximation of meromorphic functions.