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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!

Sriram
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  • I don't believe the formula for $\phi(t)$ is correct as it seems to be derived from the formula $\xi(s)=-\pi^{\frac{s-1}{2}} s\ \Gamma\left(\frac{3}{2}-\frac{s}{2}\right) \sum\limits_{n=1}^\infty n^{s-1}$ which I believe is only valid for $\Re(s)<0$. – Steven Clark Jul 28 '22 at 22:23
  • The formula is correct. It is in Titchmarsh p.254 Eq.10.1.4 – Sriram Jul 28 '22 at 23:31
  • Mathematica indicates $2 \pi \mathcal{F}t^{-1}\left\left(4 \pi ^2 n^4 e^{\frac{9 t}{2}}-6 \pi n^2 e^{\frac{5 t}{2}}\right) e^{\pi \left(-n^2\right) e^{2 t}}\right=\frac{1}{2} \pi ^{-\frac{1}{4}+\frac{i u}{2}} (-1-2 i u)\ \Gamma\left(\frac{5}{4}-\frac{i u}{2}\right) n^{-\frac{1}{2}+i u}$ which is the inverse Fourier transform of the $\phi(t)$ term. Substituting $u\to -i \left(s-\frac{1}{2}\right)$ and summing over $n$ leads to $\xi(s)=-\pi^{\frac{s-1}{2}} s\ \Gamma\left(\frac{3}{2}-\frac{s}{2}\right) \sum\limits{n=1}^\infty n^{s-1}$ which converges for $\Re(s)<0$. – Steven Clark Jul 29 '22 at 00:15
  • Can anyone comment on this question? Do 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,∞] ? If so, what are those properties? – Sriram Jul 29 '22 at 00:28
  • The Fourier transform of a Schwartz function is also a Schwartz function (see https://mathworld.wolfram.com/SchwartzFunction.html), but I'm not sure if there's a more general result. The linked paper and your question both reference the Fourier transform, whereas Titchmarsh is based on the Fourier cosine transform. $\phi(t)$ may be an even function but the sum term of $\phi(t)$ is not, and hence the term-wise Fourier transform and term-wise Fourier cosine transform of $\phi(t)$ are not equivalent. – Steven Clark Jul 29 '22 at 18:30

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