So I'm concerned showing $k^2 e^{ikx} \rightharpoonup 0$ in the sense of distributions or in other words for any $\phi \in C_c^{\infty}(\mathbb{R})$ we have for any $\epsilon > 0$ $$ \left\lvert \int_{-l}^l k^2 e^{ikx} \phi(x) \, \mathrm{d} x \right\rvert < \epsilon $$ for sufficiently large $k$ and where $l$ is defined so that $\operatorname{sup}(\phi) \subset \subset [-l, l]$
I've tried a few things:
- Change of variables $u = kx$ and $u = ikx$
- Integration by parts so as to get rid of the $k^2$ weight (I end up with something like $-\int_{-l}^l e^{ikx} \phi''(x) \, \mathrm{d} x$ and have no idea to go from there).
- Fourier analysis; I tried to use the fact that $e^{ikx}$ forms an orthonormal basis in $L^2[-\pi, \pi]$ but it would appear this won't work since we don't know that $l < \pi$ (in other words $\phi$ may not be in $L^2[-\pi, \pi]$)
- Hilbert space methods; in other words I tried to use the fact that an orthonormal basis weakly converges to $0$, but I ran into similar (if not exactly the same) issues as with Fourier analysis (also the fact that there is a $k^2$ out in front)
I'm open to hints more so than full on solutions. To be honest I don't quite have an intuition as to why this is true, but when reading on distributions it was said to be an exercise to show that it is true (thus leading me to believe it is true).
I am putting Fourier analysis in the tags due to the fact that it looks awful Fourier-y...
You say you get to $$ -\int_{-l}^l e^{ikx} \phi''(x) \, \mathrm{d} x $$ but then apply the Riemann-Lebesgue lemma.