Carathéodory–Julia type conditions and symmetries of boundary asymptotics for analytic functions on the unit disk
✍ Scribed by Vladimir Bolotnikov; Alexander Kheifets
- Publisher
- John Wiley and Sons
- Year
- 2009
- Tongue
- English
- Weight
- 248 KB
- Volume
- 282
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
Abstract
It is shown that the following conditions are equivalent for the generalized Schur class functions at a boundary point t~0~ ∈ 𝕋: 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 𝔻 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)