Can zeros of the Fourier transform of an asymmetric kernel all be real?

Question

Can zeros of the Fourier transform of an asymmetric kernel all be real ?

Be specific, in the following, if $f(t)$ is a real function, NOT an EVEN function, $z$ is complex,

${\displaystyle {\hat {f}}(z )=\int _{-\infty }^{\infty }f(t)\ e^{-2\pi itz }\,dt}$

can the zeros of ${\hat {f}}(z )$ all be real ?

Background for asking the question:

I read somewhere, it says:

${\displaystyle Theorem }$: the Fourier transform of an asymmetric kernel $f(t)$ will have infinite non-real zeros.

Can anyone give a reference of the proof of this theorem ? Of if it simple, give a proof ?

Thank you.

Answer 1

The Fourier transform of the odd real function $$ f(t)=e^{-\pi^2t^2}\sinh(\pi t) $$ is $$ \hat{f}(z)=-ie^{1/4}\pi^{-1/2} \sin z\,e^{-z^2} $$ and its zeroes are the zeroes of $\sin z$, so are all real.