To start, the issue of notation: $ \mathbb{T} $ in this case refers to $ \mathbb{R} / 2 \pi \mathbb{Z} $, or the interval $ \left[-\pi, \pi \right] $ with its endpoints identified. Given a function $ f $ on this domain, $ \hat{f} (n) $ refers to the $ n $-th Fourier coefficient $ \frac{1}{2 \pi} \int_{\mathbb{T}} f(x) e^{-inx} \mathrm{d} x $. Finally, $ S_n f $ refers to the partial Fourier sum $ \sum_{k = -n}^n \hat{f} (n) e^{inx} $. Throughout we will assume that $ f \in C^0 (\mathbb{T}) $.
Now, first suppose that $ \sum_{n \in \mathbb{Z}} \vert \hat{f} (n) \vert < \infty $. I want to show that $ S_n f \rightarrow f $ uniformly. It is immediately clear that $ S_n f $ converges uniformly, using the Weierstrass M-test. My first question is why must the limit be $ f $ (of course, it suffices to show that this is the pointwise limit everywhere). We've done a number of convergence results so far; e.g. two versions of Dini's test, Lipschitz continuity implies p.w. convergence and Jordan's Criterion, but none of those seem to help. I imagine that the answer here is very basic but I cannot seem to figure it out myself.
Secondly, suppose that $ \sum_{n \in \mathbb{Z}} \vert n \hat{f} (n) \vert < \infty $. I want to show that $ f \in C^1 (\mathbb{T}) $. Clearly this condition is stronger than the first and so $ S_n f \rightarrow f $. Of course, our hypothesis seems to imply a form of termwise differentiation, but I know we can't in general do that like we can for power series (for instance, $ x \rightarrow \vert x \vert $ satisfies the first condition but ought not to be differentiable at $ 0 $). I tried doing this from first principles but it seems like a dead end.
I should add that I would much prefer good hints to clear answers, although the first part may well be so trivial that a hint is as good as an answer.
Let $g(t)=\sum \hat {f} (n) e^{int}$. Then (by uniform convergence) $\hat {g} (k)=\frac 1 {2\pi} \int g(x)e^{-ikx}\, dx=\sum _n \frac 1 {2\pi} \int \hat {f} (n)e^{inx-ikx}\, dx$ which gives $\hat {f} (k)=\hat {g} (k)$ for all $k$. This implies that $f=g$ so the sum of the Fourier series is exactly $f$. For the second part consider the series $\sum_n in \hat {f} (n) e^{inx}$. This series converges uniformly. Now use the following basic analysis result:
Lemma If $S_n$ is a sequence of $C^{1}$ functions such that $S_n \to f$ uniformly and $S_n' \to g$ uniformly then $f$ is differentiable and $f'=g$.