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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Holomorphic perturbation of Fourier coefficients


Author: Thomas Vils Pedersen
Journal: Proc. Amer. Math. Soc. 129 (2001), 1365-1366
MSC (2000): Primary 42A16; Secondary 46J20
Published electronically: October 11, 2000
MathSciNet review: 1814161
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Abstract: Let $\mathbb{T}$ be the unit circle, let $\mathcal{B}$ be a Banach space continuously embedded in $L^1(\mathbb{T})$ and suppose that $\mathcal{B}$ is a Banach $L^1(\mathbb{T})$-module under convolution. We show that if $f(z)=\sum_{n=-\infty}^{\infty} a_nz^n\in\mathcal{B}$ and $F$ is holomorphic in a neighbourhood $U$ of $0$ with $F(0)=0$ and $a_n\in U (n\in\mathbb{Z}),$ then $\sum_{n=-\infty}^{\infty} F(a_n)z^n\in\mathcal{B}.$


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Additional Information

Thomas Vils Pedersen
Affiliation: Laboratoire de Mathématiques Pures, Université Bordeaux 1, 351, cours de la Libération, F-33405 Talence cédex, France
Email: vils@math.u-bordeaux.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05785-3
PII: S 0002-9939(00)05785-3
Received by editor(s): July 20, 1999
Published electronically: October 11, 2000
Additional Notes: This work was carried out at Université Bordeaux 1 while the author was holding a TMR Marie Curie postdoctoral grant from the European Commission.
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society