Let $k(x,y)=\exp(-(x-y)^2/2)$ be a gaussian kernel and $f(x)=\exp(-x^2/2)\sin(6x)$ to be a function, I'd like to compute the RKHS norm for f: $$\Vert f\Vert^2_H=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\mid \tilde{f}(w)\mid^2 \exp(w^2/2)dw$$
where $\tilde{f}(w):=\frac{1}{\sqrt{2\pi}}\int f(x) \exp(-iwx)dx$
I've failed to brute force it so I'm hoping that there's some trick to do it.