The following "twisted" Poisson Summation formula for $\chi$ primitive of conductor $q$ :
$$ \sum_{n\in\mathbb{Z}}\chi(n)f\left(\frac{nx}{\sqrt{q}}\right) = \frac{A}{x}\sum_{n\in\mathbb{Z}}\bar\chi(n)\hat f\left(\frac{n/x}{\sqrt{q}}\right) $$
holds for integrable continuous functions $f:\mathbb{R}\to\mathbb{C}$ satisfying $|f(t)|+|\hat f(t)|\ll (1+|t|)^{-1-\delta}$ for some $\delta>0$
but I think this holds also for functions having a singulartiy in zero providing they have a Fourier transform and a good behavior at infinity like: $f(x)=|x|^{-\frac{1}{2}} e^{-x^2} $
I did not find any reference on this subject, Poisson summation formula is always shown for functions satisfying $|f(t)|+|\hat f(t)|\ll (1+|t|)^{-1-\delta}$ for some $\delta>0$.
Does it exists somewhere ? Or I will need to make the demo ? (I would prefer a reference!)
(Here $A:=\sqrt{q}/\tau(\bar\chi)$ is the so-called root number, it is of modulus $1$)
