Consider a continuous function $f: S^1 \to \mathbb{C}$. Suppose we can write $$f(z) = \sum_{n \in \mathbb{Z}} a_n z^n , \quad z \in S^1$$
where $(a_n)_n \in l^1(\mathbb{Z})$. Is the sequence $(a_n)_n$ unique? I.e. if $(a_n)_n, (b_n)_n \in l^1(\mathbb{Z})$, do we have
$$\sum_{n \in \mathbb{Z}}a_n z^n = \sum_{n \in \mathbb{Z}} b_n z^n \implies a_n = b_n \quad \forall n \in \mathbb{Z}$$
I couldn't find a reference online.