Constructive Approximation

, Volume 34, Issue 3, pp 393–420

Growth Behavior and Zero Distribution of Rational Approximants

Article

Abstract

We investigate the growth and the distribution of zeros of rational uniform approximations with numerator degree ≤n and denominator degree ≤mn for meromorphic functions f on a compact set E of ℂ where mn=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 Eρ(f) of meromorphy of f if f has a singularity of multivalued character on the boundary of Eρ(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.

Keywords

Rational approximation Distribution of zeros Jentzsch–Szegő-type theorems Padé approximation m1-maximal convergence Harmonic majorants 

Mathematics Subject Classification (2010)

41A20 26C15 30E10 41A21 41A25 

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Mathematisch-Geographische Fakultät, Lehrstuhl für Mathematik—Angewandte MathematikKatholische Universität Eichstätt-IngolstadtEichstättGermany
  2. 2.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria

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