It is well-known that the map $T:L^1(\mathbb{T})\rightarrow c_0(\mathbb{Z})$, given by $Tf=\{\hat{f}(n)\}_{n\in\mathbb{Z}}$, where $\hat{f}(n)$ is the $n$-th Fourier coefficient of $f$, is a linear, bounded and injective map but not a surjective map. I'm interested in finding a double sequence $\{c_n\}_{n\in\mathbb{Z}}\in c_0(\mathbb{Z})$ such that it doesn't have an inverse image by $T$ in $L^1(\mathbb{T})$. Basically, a counterexample of the surjectivity of $T$.
Can you give me an idea of how to construct such a sequence or a reference of where I can find it?
Thank you very much for your time!