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Holomorphic perturbation of Fourier coefficients

Authors:	Thomas Vils Pedersen

Institution:	Laboratoire de Mathématiques Pures, Université Bordeaux 1, 351, cours de la Libération, F-33405 Talence cédex, France

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 is holomorphic in a neighbourhood of with 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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