Holomorphic perturbation of Fourier coefficients

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 $\sum _{n=-\infty }^{\infty } F(a_n)z^n\in \mathcal {B}.$