One should prove the following:
$\mathcal{F}_k^{-1}[\tilde f (k)\tilde g (k)](x)=\frac{1}{\sqrt{2\pi}}(f * g)(x)$
I have absolutely no idea how to solve this exercise. I researched for articles and websites regarding this topic, but couldn't find anything similar :( I would be very thankful for any help!
$$\begin{align} \int (f*g)(x) \, e^{-ix\xi} \, dx &= \int \left( \int f(x-y)\,g(y)\,dy \right) \, e^{-ix\xi} \, dx \\ &= \iint f(x-y)\,g(y)\,e^{-ix\xi}\,dy\,dx \\ &= \int \left( \int f(x-y) \, e^{-ix\xi} \, dx \right) g(y) \, dy \\ &\stackrel{z:=x-y}{=} \int \left( \int f(z) \, e^{-i(y+z)\xi} \, dz \right) g(y) \, dy \\ &= \int \left( \int f(z) \, e^{-iz\xi} \, dz \right) g(y) e^{-iy\xi} \, dy \\ &= \left( \int f(z) \, e^{-iz\xi} \, dz \right) \left( \int g(y) e^{-iy\xi} \, dy \right)\end{align} $$