Function with positive Fourier coefficients decaying as $\frac{1}{n}$ Is there an example (I'm not looking for a sufficient or necessary condition but just for an example) of a bounded Rieman-integrable function $f\colon [-\pi,\pi]\rightarrow\mathcal{R}$ with Fourier coefficients 
$c_n(f) = \int_{-\pi}^\pi{f(t)e^{-int}\,dt}$ that are positive and decay as $\frac{1}{n}$ to zero (so that the Fourier coefficients are not summable)?
From this discussion a sufficient condition would be to find a function which has Fourier coefficients decaying as $n^{-1/2}$ and I could then take the convolution of this function with itself to obtain what I want.

No, there is no such function.


Suppose $f(t)\sim\sum_nc_ne^{int}$ where $c_n\ge0$ and $\sum c_n=\infty$. Then $f$ is not bounded (hence not Riemann integrable).


Not-quite proof: Let $t=0$: it follows that $f(0)=+\infty$.
Of course that's not quite a proof, since there's no reason a priori that the series should converge to $f(0)$ for $t=0$. It does make it clear that the answer must be no, and if you have any feeling for real analysis it seems clear that the condition $c_n\ge0$ means that it can't be hard to fix the argument.
Hint for an actual proof: Show that if $f$ is bounded then the Fejer means (or the Abel means) of the Fourier series must be uniformly bounded.
Edit: It appears that the hint was not sufficient. The argument is very simple. First, if $f$ is bounded then the Fejer means are uniformly bounded, as hinted: $\sigma_n=f*K_n$, so $$||\sigma_n||_\infty\le||f||_\infty||K_n||_1=||f||_\infty.$$
On the other hand, it's obvious that if $c_n\ge0$ and $\sum c_n=\infty$ then the Fejer means are not uniformly bounded:
$$\sigma_n(0)=\sum_j(1-|j|/n)^+c_j\ge\frac12\sum_{|j|<n/2}c_j.$$
