Show $\sum_{n \in \mathbb Z} |\hat f(n)| \leq \frac{1}{\sqrt{2\pi}}||f||_1 + \frac{\pi}{\sqrt 3}||f'||_2$ for $f,f' \in L_\mathbb C ^2 [0. 2\pi] $

Question

Let $f,f' \in L_\mathbb C ^2 [0. 2\pi] $.

Show $$\sum_{n \in \mathbb Z} |\hat f(n)| \leq \frac{1}{\sqrt{2\pi}}||f||_1 + \frac{\pi}{\sqrt 3}||f'||_2$$

What I have tried:

By Integration by parts, we have $\hat {f'}(n)=in \hat f(n)$ for $n \neq 0$. I know by Plancherel theorem, $||f||_2 = ||\hat f ||_{\ell^2}$ and $||f'||_2 = ||\hat {f'} ||_{\ell^2}$.

How can I proceed? Any help is appreciated.

Answer 1

$|\hat f(0)|\le\frac1{\sqrt{2\pi}}||f||_1$. And $$\sum_{n\ne0}|\hat f(n)| =\sum\frac1{|n|}|\widehat{f'}(n)|\le\left(\sum_{n\ne0}\frac1{n^2}\right)^{1/2} \left(\sum|\widehat{f'}(n)|^2\right)^{1/2}...$$