Constructive Approximation

, Volume 34, Issue 3, pp 393–420

Growth Behavior and Zero Distribution of Rational Approximants

Authors

    • Mathematisch-Geographische Fakultät, Lehrstuhl für Mathematik—Angewandte MathematikKatholische Universität Eichstätt-Ingolstadt
  • Ralitza K. Kovacheva
    • Institute of Mathematics and InformaticsBulgarian Academy of Sciences
Article

DOI: 10.1007/s00365-010-9124-5

Cite this article as:
Blatt, H. & Kovacheva, R.K. Constr Approx (2011) 34: 393. doi:10.1007/s00365-010-9124-5

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 approximationDistribution of zerosJentzsch–Szegő-type theoremsPadé approximationm1-maximal convergenceHarmonic majorants

Mathematics Subject Classification (2010)

41A2026C1530E1041A2141A25

Copyright information

© Springer Science+Business Media, LLC 2010