Limit of Cosine Integral

Question

Let $f\in L^1(\mathbb{R})$. Prove that $$\lim_{|\xi|\to\infty}\int_{\mathbb{R}}f(x)\cos(\xi x)dx=0$$

Answer 1

Hint

Let $\varepsilon>0$. By density of step function in $L^1$, there is $g$ a step function s.t. $\displaystyle\int|f-g|<\varepsilon.$

You have that $$\left|\int f(x)\cos(x\xi)dx\right|\leq \varepsilon+\left|\int g(x) \cos(x\xi)dx\right|$$

To show that $$\lim_{|\xi|\to \infty }\int g(x)\cos(x\xi)dx=0,$$ is easy using the fact that $g$ is a step function. The claim follow.