I have to prove the following:
$$||gu||_2 \leq ||u||_2$$
where $g(w)=(2n^2/\omega^2)(1-cos(\omega/n)), n \in \mathbb{N}$, and $u(\omega)$ a general function $\in$ $L^2$.
Can I state the following?
$$\int_{-\infty}^{+\infty}|(2n^2/\omega^2)(1-cos(\omega/n))u(\omega)|^2d\omega \leq \int_{-\infty}^{+\infty}|(2n^2/\omega^2)u(\omega)|^2d\omega \leq \int_{-\infty}^{+\infty}|u(\omega)|^2d\omega$$
The first passage is pretty obvious because $|(1-cos(\omega/n))| \leq 1$, but the second one is actually true when $\omega \rightarrow \infty$. If not, how could I prove it?
$1-\cos t \leq \frac {t^{2}} 2$. This gives the desired inequality. In your argument both steps are wrong. $1-\cos t$ can take the value $2$ so the first step is wrong. The second step is also wrong.