Premium
Carathéodory–Julia type conditions and symmetries of boundary asymptotics for analytic functions on the unit disk
Author(s) -
Bolotnikov Vladimir,
Kheifets Alexander
Publication year - 2009
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200610809
Subject(s) - mathematics , unit disk , analytic function , analytic continuation , homogeneous space , type (biology) , boundary (topology) , mathematical analysis , order (exchange) , function (biology) , class (philosophy) , hermitian matrix , isometry (riemannian geometry) , pure mathematics , unit (ring theory) , continuation , reflection (computer programming) , boundary value problem , geometry , ecology , mathematics education , finance , evolutionary biology , artificial intelligence , computer science , economics , biology , programming language
It is shown that the following conditions are equivalent for the generalized Schur class functions at a boundary point t 0 ∈ : 1) Carathéodory–Julia type condition of order n ; 2) agreeing of asymptotics of the original function from inside and of its continuation by reflection from outside of the unit disk up to order 2 n + 1; 3) t 0 ‐isometry of the coefficients ofthe boundary asymptotics; 4) a certain structured matrix ℙ constructed from these coefficients being Hermitian. It is also shown that for an arbitrary analytic function, properties 2), 3), 4) are still equivalent to each other and imply 1) (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)