I have a function of the form
$$f(x)=e^{-tx^2/2}g(x)$$
where $g(x+T)=g(x)$ is a periodic function.
I also know that its Fourier transform can be written as
$$\hat f(r)=e^{-tr^2/2}h(r)$$
for another periodic function $h(r+T/2)=h(r)$ with half the period of the the function $g$.
I suspect that there exists a relationship between $g(x)$, $\hat h(x)$ but I am not sure how to find it.
The two periodic functions can be related using
$$h(r)=e^{tr^2/2} [\widehat{e^{-tx^2/2}}*\hat g(x)](r)$$
Then, by writing the two functions as Fourier series
$$g(x)=\sum_k a_k e^{2\pi i kx/T} \quad h(r)=\sum_m b_m e^{4\pi i mr/T}$$
we can find
$$\sum_k b_k e^{4\pi i kr/T}=\frac{\sqrt{2\pi}}{\sqrt{t}}e^{tr^2/2}\sum_k a_k [e^{-r^2/2t}*\widehat{e^{2\pi i kx/T}}]=\frac{\sqrt{2\pi}}{\sqrt{t}}e^{tr^2/2}\sum_k a_k [e^{-r^2/2t}*\delta(r-kT)]$$
which can be reduced to
$$\sum_k b_k e^{4\pi i kr/T}=\frac{\sqrt{2\pi}}{\sqrt{t}}e^{tr^2/2}\sum_k a_k e^{-(r-kT)^2/2t}$$.
we can write the relationship between the two series as $$\sum_m b_m e^{4\pi i mr/T}=\frac{\sqrt{2\pi}}{\sqrt{t}}e^{tr^2/2}\sum_k a_k e^{-(x-kT)^2/2t}$$
which looks a lot like the Poisson summation formula. Is there a way to find a relationship between $g$ and $\hat h$? Or a method to evaluate $a_k$ $b_m$ in terms of each other?