Properties of Fourier transforms on the positive real axis

Question

Consider $\hat{f}(\omega)$, the Fourier transform of a function $f(t)$. Let $f(t)$ be real, positive, and only non-zero for $t>0$. With these assumptions, the function $\hat{f}(\omega) \hat{f}(-\omega)$ is real (this part is easy to see, and just follows from $f$ being real) and appears to always monotonically decrease for $\omega > 0$ (less easy to see, this only comes from me checking many different functions and seeing that they all satisfy this). Is this second statement a general fact? It seems you can get somewhere by noting that $\hat{f}'(\omega) = i \hat{g}(\omega)$, where $\hat{g}$ is the Fourier transform of a real and positive function (positivity comes from the fact that $f(t)$ is non-zero for $x>0$), but I'm not sure where to go from there.

Answer 1

If $f$ satisfies your properties let $g(t)=f(t)+f(t-2)$ what is $\hat{g},|\hat{g}|^2$ in term of $\hat{f}$