Let $f$ be a $2\pi$-periodic continuously differentiable function. Let $\hat{f}(n)$ denote the $n$-th Fourier coefficient of $f$. How would I prove that $$\sum_{n=-\infty}^\infty \left | \hat{f}(n) \right |$$ converges?
I have looked on StackExchange for the same question, but I all could find were half-solutions. From what I can tell, I have to use Parseval's identity and the Cauchy-Schwartz inequality on $f'$, but it is unclear to me how/where to use the C-S inequality.
I would greatly appreciate it if someone could provide a step-by-step solution.
Best Answer
You can show that $\hat{f_k'} = i k \hat{f_k}$ and $\sum_k |\hat{f_k'}|^2 < \infty $.
Cauchy Schwartz gives:
$\sum_k |\hat{f_k}| = |f_0| + \sum_{k \neq 0} |\hat{f_k}| \le |f_0| + \sum_{k \neq 0} {1 \over |k|}|k\hat{f_k}| \le |f_0|+\sqrt{\sum_{k \neq 0} {1 \over k^2} } \sqrt{ \sum_k |\hat{f_k'}|^2 }$.