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.



Fourier series representing a continuous function?
Maybe you are already aware of the following if-and-only-if condition, though I wouldn't say it counts as "easily checkable": let $(a\_k)$ be the coefficient sequence, let $s\_N$ denote the partial sums
$$
s\_N(z) = \sum\_{k=-N}^N a\_k z^k
$$
and let $\sigma\_N(z)$ denote the Cesaro means 
$$
\sigma\_N(z)=\frac{1}{N+1}\sum\_{n=0}^N s\_N(z).
$$
Then $(a\_k)\_{k\in\mathbb Z}$ is the Fourier series of a continuous function on the circle if and only if the sequence $\sigma\_N$ is uniformly Cauchy on the circle, in which case it converges to a continuous function $f$ with $\hat{f}(k)=a\_k$. (The sufficiency of this condition is clear from term-by-term integration, the necessity is Fejer's theorem.)


