I am struggle to prove the following theorem.
If $f$ $\in$ $C^1$ and $f$ , $f'$ $\in$ $L^1$ , then $\mathcal{F}$$f$ $\in$$L^1$.
($\mathcal{F}$$f$ is Fourier transform of $f$ )
I doubt that whether this theorem is right.
Please tell me proof or counterexample.
(following statement is supplement)
I already know that $f$ is bounded , $\mathcal{F}$$f$ $\in$ $L^2$ and $f$ $\in$ $L^2$ under the hypothesis.
And, since $f$, $f'$ $\in$ $L^1$ then $\mathcal{F}$$f$ , $\mathcal{F}$$f'$ is bounded.