, Volume 34, Issue 3, pp 393-420
Date: 27 Oct 2010

Growth Behavior and Zero Distribution of Rational Approximants

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Abstract

We investigate the growth and the distribution of zeros of rational uniform approximations with numerator degree ≤n and denominator degree ≤m n for meromorphic functions f on a compact set E of ℂ where m n =o(n/log n) as n→∞. We obtain a Jentzsch–Szegő type result, i.e., the zero distribution converges weakly to the equilibrium distribution of the maximal Green domain E ρ(f) of meromorphy of f if f has a singularity of multivalued character on the boundary of E ρ(f). The paper extends results for polynomial approximation and rational approximation with fixed degree of the denominator. As applications, Padé approximation and real rational best approximants are considered.

Communicated by Doron S. Lubinsky.
This work was supported by DFG-Research Grant (Germany) BL 272/10-1.