I've been trying to re-express the particular theta function $$\theta(r)=\frac{1}{2}\sum_{n\in\mathbb{Z}}\cos(nr)\exp(-n^2/\lambda)$$ in the form $$\theta(r)=\sum_{n\in\mathbb{Z}}\alpha_n\exp(-\lambda(r-2\pi n)^2/\beta)$$ for some $\alpha_n$ and $\beta$. It seems to me that it should be possible, based on intuition from considering diffusion over $S^1$: the $\theta$ function is a fundamental solution of such a system, but it also seems we should be able to take the fundamental solution over $\mathbb{R}$ and sum over the concentrations at $\{r+2\pi n : n\in\mathbb{Z}\}$ to arrive at the same solution.
I've considered writing the cosine term in terms of complex exponentials, for which I get $$\theta(r)=\sum_{n\in\mathbb{Z}}\exp(inr-n^2/\lambda),$$ but from here I don't see how to proceed.
Would greatly appreciate any help! If my assumption is incorrect, would love to hear where my logic is lacking.
Thank you, @reuns! Here's my solution:
For a Fourier transform $S$ of a function $s$ defined by $\displaystyle S(f)\triangleq\int_\mathbb{R}s(x)e^{-i2\pi fx}dx$, we have that $$s_{2\pi}=\sum_{n\in\mathbb{Z}}\frac{1}{2\pi}S\left(\frac{k}{2\pi}\right)e^{ikx},$$ Eqn. 2 here, where $\displaystyle s_{2\pi}(x)\triangleq\sum_{n\in\mathbb{Z}}s(x+2\pi n).$ We have that the Fourier transform of the Gaussian $s(x)=e^{-\lambda x^2}$ is $$S(f) = \sqrt{\frac{\pi}{\lambda}}e^{-(\pi f)^2/\lambda},$$ and so we have that $$\sum_{n\in\mathbb{Z}}\exp(-\lambda(r+2\pi n)^2)=\sum_{k\in \mathbb{Z}}\frac{1}{2\sqrt{\pi\lambda}}\exp\left(-\frac{1}{4\lambda}k^2\right)e^{ikr},$$ which reduces to $$\sum_{n\in\mathbb{Z}}\exp(-\lambda(r+2\pi n)^2)=\frac{1}{2\sqrt{\pi\lambda}}\sum_{k\in \mathbb{Z}}\cos(kr)\exp\left(-\frac{1}{4\lambda}k^2\right).$$ Finally, here is a Desmos plot.