Overconvergence of Rational Approximants of Meromorphic Functions

Abstract

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 E _ρ(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 Padé approximation can be derived. Hereby, well-known results for polynomial approximation of holomorphic functions are generalized for rational approximation of meromorphic functions.

Dedicated to the memory of Q. I. Rahman