How to use hint $\hat{f}(x)=-\frac{1}{i \xi}\int i \xi e^{-i x \xi}f(x)dx$ to show Riemann Lebesgue Lemma

Question

Let $f \in C_{c}^{\infty}(\mathbb R)$ and I have proven that:

$\vert \vert \hat{f} \vert \vert_{\infty}\leq\vert \vert f\vert\vert_{1}, (1)$

Then I am told to use the hint that:

$\hat{f}(\xi)=-\frac{1}{i \xi}\int (-i \xi) e^{-i \xi x}f(x)dx$

$\lim\limits_{\vert \xi\vert \to \infty}\hat{f}(\xi)=0$

I can see that I need find an appropriate bound and $(1)$ seems useful and then I'd just let $\vert \xi\vert \to \infty$ but I am struggling to find the upper bound in terms of $\vert \vert f\vert\vert_{1}$. How am I supposed to treat the term $(-i \xi) e^{-i \xi x}$?

Answer 1

They want you to integrate by parts. Try to see if that helps; I've added a spoiler for you below, if you need it.

Let $\xi\neq 0.$ Integrating by parts, we get that $$\hat{f}(\xi)=\frac{1}{i\xi}\hat{f_x}(\xi),$$ where the boundary term vanishes due to the fact that $f$ is compactly supported. So, for any $\xi\neq 0$, we have $$|\hat{f}(\xi)|\leq \frac{1}{|\xi|}\|f_x\|_{L^1},$$ by your estimate. Now, you can just take the limit to conclude.