Finite Blaschke products and the construction of rational Γ-inner functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:447

Article

Agler, Jim¹  Lykova, Zinaida A.²  Yaung, N. J.^2,3 

[1] Univ Calif San Diego, Dept Math, San Diego, CA 92103 USA

[2] Newcastle Univ, Sch Math & Stat, Newcastle Upon Tyne NE1 7RU, Tyne & Wear, England

[3] Univ Leeds, Sch Math, Leeds LS2 9JT, W Yorkshire, England

关键词: Blaschke product;    Symmetrized bidisc;    Interpolation;    Pick matrix;    Complex geodesic;   

DOI  :  10.1016/j.jmaa.2016.10.035

来源: Elsevier

【 摘 要 】

A Gamma-inner function is a holomorphic map h from the unit disc D to Gamma whose boundary values at almost all points of the unit circle T belong to the distinguished boundary b Gamma of Gamma. A rational Gamma-inner function h induces a continuous map h vertical bar(T) from T to b Gamma. The latter set is topologically a Mobius band and so has fundamental group Z. The degree of h is defined to be the topological degree of h vertical bar(T). In a previous paper the authors showed that if h = (s, p) is a rational Gamma-inner function of degree n then s(2) - 4p has exactly n zeros in the closed unit disc D-, counted with an appropriate notion of multiplicity. In this paper, with the aid of a solution of an interpolation problem for finite Blaschke products, we explicitly construct the rational Gamma-inner functions of degree n with the n zeros of s(2) - 4p prescribed. (C) 2016 The Authors. Published by Elsevier Inc.

【 授权许可】

Free