This is maybe not really research level, but I have not found anything in the literature, and asking on math.stackexchange wasn't successful either.

Fourier series define an isometry $L^2(\mathbb{Z}) \rightarrow L^2(S^1), (a_k)_{k \in \mathbb{Z}} \mapsto \sum a_k z^k \colon S^1 \rightarrow \mathbb{C}$ (all Hilbert spaces are complex). It is known (and easy) that, when we start with a sequence $a_k$ which is actually in $L^1(\mathbb{Z})$, we end up with a continuous function on $S^1$. My question is: What more can we say about sequences $a_k$ whose associated Fourier series is a continuous function on $S^1$? In particular, is there an easily checkable if and only if-criterion? I would guess that not, but I have never seen such a question discussed in the literature.