Consider a Gaussian function $f(t) = e^{- \pi t^2}$ which is strictly decreasing from $t=[0, \infty]$. It is a Fourier transformable, positive and even symmetric analytic function.
Do such strictly decreasing functions have specific properties in Fourier transform(FT) domain? May be their asymptotic fall-off rates? Can we argue that if FT of a function f(t) has specific properties, then f(t) should be strictly decreasing from $t=[0, \infty]$ ? If so, what are those properties?
I am trying to show that the Inverse Fourier transform of Riemann's Xi function on the critical line, given by $\phi(t)$ is strictly decreasing from $t=[0, \infty]$. $\phi(t)$ is an even symmetric and positive analytic function.
$\phi(t) = \sum_{n=1}^{\infty} [ 4 n^{4} \pi^{2} e^{\frac{9t}{2}} - 6 n^{2} \pi e^{\frac{5t}{2}} ] e^{- \pi n^{2} e^{2t}}$
Brian Conrey states this result in Page 5 in his article, but no proof available. https://www.ams.org/notices/200303/fea-conrey-web.pdf#page=5
For Gaussian function $f(t) = e^{- \pi t^2}$, we can easily show that df(t)/dt < 0, for t>0.
For $\phi(t)$, it is hard to show this result, due to infinite summation.
A similar question was posted earlier in another math website, but answers didn't help.
Thanks in advance!