I was trying to calculate $f * g$ where $f(t) = F(\omega) e^{-i \omega t}$ and $g(t) = G(\omega) e^{-i \omega t}$. I did are the following.
$$(f*g)(t)=\int^\infty_{-\infty} F(\omega)e^{-i \omega (t- \tau)}G(\omega)e^{-i \omega \tau} d\tau=F(\omega)G(\omega)e^{( \omega t)}\int^\infty_{-\infty}d\tau=\infty$$
Is it correct? I was expecting something like
$$(f * g) (t) = F(\omega)G(\omega)e^{( \omega t)}$$