I am trying to show that the Fourier transform of $\cosh(x)^{-\nu}$ is positive for every $\nu>1$. I know that such a function has even Fourier transform and... that's about it. Could you suggets a way to attack this problem?
Thanks.
PS: the positivity is ensured by some piece of advanced theory, but I would like to prove it by getting my hands "dirty", although it seems rather difficult.