Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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?)

  • [E] S. W. Ellacott, On the Faber transform and efficient numerical rational approximation, SIAM J. Numer. Anal. 20 (1983), 989-1000. MR 714694 (85f:41010)
  • [G] D. Gaier, Vorlesungen über Approximation im Komplexen, Birkhäuser, Basel, 1980. MR 604011 (82i:30055)
  • [P] C. Pommerenke, Univalent functions, Vandenhoeck and Ruprecht, Göttingen, 1975. MR 0507768 (58:22526)
  • [R] W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1966. MR 0210528 (35:1420)
  • [S] G. Schoeber, Univalent functions, Lecture Notes in Math., vol. 478, Springer-Verlag, Berlin and New York, 1975.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30B40, 30C99

Retrieve articles in all journals with MSC: 30B40, 30C99


Additional Information

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

American Mathematical Society