Holomorphic perturbation of Fourier coefficients

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