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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The Faber transform and analytic continuation


Author: Elgin Johnston
Journal: Proc. Amer. Math. Soc. 103 (1988), 237-243
MSC: Primary 30B40; Secondary 30C99
DOI: https://doi.org/10.1090/S0002-9939-1988-0938675-5
MathSciNet review: 938675
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Abstract: Let $ \Omega \subseteq C$ be a bounded, simply connected domain, and let $ \left\{ {{\Phi _n}\left( w \right)} \right\}_{n = 0}^\infty $ be the Faber polynomials associated with $ \Omega $. Given $ f\left( z \right) = \sum\nolimits_{k = 0}^\infty {{c_k}{z^k}} $ analytic in $ \Delta \left( {0,1} \right)$ we consider the function

$\displaystyle F\left( w \right) = \sum\limits_{k = 0}^\infty {{c_k}{\Phi _k}\left( w \right)} .$

We show that with proper restrictions on $ \partial \Omega $, the existence of an analytic continuation of $ f$ across a subarc of $ C\left( {0,1} \right)$ implies the existence of an analytic continuation of $ F$ across a subarc of $ \partial \Omega $. Some converse results are also established.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0938675-5
Article copyright: © Copyright 1988 American Mathematical Society