Suppose $f$ is absolutely continuous and the Fourier coefficients of $f$ satisfy $ \hat f(n)= \frac{a_n sgn(n)}{n} \ge 0 $
where the $a_n$ are postive, even, and decreasing to zero as $|n| \to \infty$, so that the Fourier coefficients of $f$ are even and positive. Show $ \sum_{n=-\infty}^{\infty} \hat f(n)<\infty $
This was answered in this post. The post says:
A standard theorem says that the Fourier series of an absolutely continuous function converges to it uniformly so taking $x=0$ you get the result.
I suspect the theorem holds almost everywhere, so I don't see why taking $x=0$ is justified.
If possible, can you point me to a proof for this theorem?
Or explain why the choice is possible.