Let $\alpha\in(0,2]$ and $$f_t(u)=\sum_{m\neq 0}\frac{1}{m^\alpha}\exp(-\|m\|^2 t^2)\exp(im\cdot u),u\in \mathbb R^d.$$ Would anyone know a closed form expression for $f_t(u)$, in all dimension $d$? I actually especially need an expansion with the two first terms as $u\to \infty$.
Here is an attempt with the Poisson summation formula. Let $\varphi_t(m)=t^{-\alpha }\|m\|^{-\alpha}\exp(-\|m^2\|t^2)$. Then we have $$f_t(u)+1=c_d t^{\alpha }\sum_{j\in \mathbb Z^d}\hat \varphi_t(u+2\pi j)=c_d t^{-d-\alpha}\sum_j\hat \varphi_1(t^{-1}(u+2\pi j))$$
The problem is that this is not valid because $\varphi_1$ is not continuous, I think there is a real problem around 0. I would already be happy for a solution in the case $d=2,\alpha=2,t=0$, i.e. equivalent of $$\sum_{m\neq 0}\frac{e^{imu}}{\|m\|^2}$$