I am fairly sure the answer to my question is "No", so this is more of an affirmation/reference request question.
Given a Fourier series $\sum\limits_{k \in \mathbb{Z}} a_k e^{kxi}$, we can view it as an $L^2$-function on $S^1$. My question is: Given the Fourier coefficients $a_k$, can we decide whether this function is actually continuous? This is so if the $a_k$ are $L^1$-summable, but this is not an "if and only if"-condition. Is there an "if and only if"-condition?
Interestingly enough, in the world of Fourier transforms, the Fourier transform (i.e, coefficient) does indicate whether the function is continuous. Specifically, if the Fourier transform
$$\hat{f}(k) = \int_{-\infty}^{\infty} dx \, f(x) \, e^{i k x}$$
behaves as $1/k^{m+1}$ for some integer $m$ as $k \to \infty$, then $f(x)$ has a discontinuity in its $m$th derivative. For example, for $\hat{f}(k) = \sin{\pi k}/(\pi k)$, then $f(x)$ has a discontinuity at $x=\pm 1$ ($m=1$). For $\hat{f}(k) = \sin^2{\pi k}/(\pi k)^2$, then $f'(x)$ has a discontinuity at $x=\pm 2$ and at $x=0$. And so on.
I suspect this translates to Fourier coefficients (i.e,, periodic functions), but I cannot prove it definitively.