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Szegő Type Asymptotics for Minimal Blaschke Products

  • A. L. Levin
  • E. B. Saff
Conference paper
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 19)

Abstract

Let μ be a positive, finite Borel measure on [0,2π). For 0 <r< 1, 0 <p< ∞, let
$$En,p(d\mu ;r): = _{{B_n}}^{\inf }\left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {{{\left| {{B_n}\left( {r{e^{i\theta }}} \right)} \right|}^p}d\mu \left( \theta \right)} } \right\}1/p,$$
where the infimum is taken over all Blaschke products of ordernhaving zeros in |z| < 1. LetB n * denote a minimal Blaschke product and letG(μ ) denote the geometric mean of the derivative of the absolutely continuous part ofμ. In the first part of the paper we present a self-contained proof of a result due to Parfenov; namelyE n,p ~ r n G(μ′)1/p as n → ∞. In the second part we describe the extension of the classical Szegő function D(z) and prove that B n * (z) ~ z n G(µ′)1/p /D(z)2/p as n → ∞, uniformly on compact subsets of the annulus r < z < 1/r. Some generalizations and applications are also discussed.

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Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • A. L. Levin
    • 1
  • E. B. Saff
    • 2
  1. 1.Department of MathematicsOpen University Max Rowe Educational CenterRamat. Aviv Tel-AvivIsrael
  2. 2.Institute for Constructive Math. Department of MathematicsUniversity of South FloridaTampaUSA

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