Let $f\in L^1([0,2\pi])$ be a $2\pi$-periodic function with $\hat f(0)=0$ and $\hat f(\vert n\vert)=-\hat f(-\vert n\vert)\geq 0$. Define $F(t)=\int_0^t f(x)dx$. I know that F iscontinuous, $2\pi$-periodic function and $\hat F(n)=\frac{\hat f(n)}{in}$ if $n\neq 0$.
I want to prove that $\sum_{n\geq 1} \frac{\hat f(n)}{n}<\infty$.
How to prove that?
The function $F$ is continuous and of bounded variation because $$ V_{a}^{b}(F) = \int_{a}^{b}|f(t)|dt. $$ Therefore the Fourier series for $F$ converges pointwise everywhere to $F$ (in fact it must converge uniformly.) In particular, $$ F(0) = \lim_N\sum_{n=-N}^{N}\hat{F}(n)=2\lim_N\sum_{n=1}^{N}\frac{\hat{f}(n)}{in}. $$