Regions of meromorphy and value distribution of geometrically converging rational functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:382

Article

Blatt, H. -P¹  Grothmann, R.¹  Kovacheva, R. K.² 

[1] Kathol Univ Eichstatt Ingolstadt, Math Geog Fak, D-85071 Eichstatt, Germany

[2] Bulgarian Acad Sci, Inst Math & Informat, BU-1113 Sofia, Bulgaria

关键词: Rational approximation;    Meromorphic functions;    Distribution of zeros and poles;    a-Values;    Pade approximation;    Picard theorem;    m(1)-Maximal convergence;    Harmonic majorant;   

DOI  :  10.1016/j.jmaa.2011.04.028

来源: Elsevier

【 摘 要 】

Let D be a region, (r(n),)(n is an element of N) a sequence of rational functions of degree at most n and let each rn have at most m poles in D, for m is an element of N fixed. We prove that if (r(n)) (n is an element of N) converges geometrically to a function f on some continuum S subset of D and if the number of zeros of r(n) in any compact subset of D is of growth o(n) as n -> infinity. then the sequence (r(n))(n is an element of N) converges nil-almost uniformly to a meromorphic function in D. This result about meromorphic continuation is used to obtain Picard-type theorems for the value distribution of m(1)-maximally convergent rational functions, especially in Pade approximation and Chebyshev rational approximation. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free