Maximal polarization for periodic configurations on the real line

We prove that among all 1-periodic configurations $\mathrm{\Gamma }$ of points on the real line $\mathbb{R}$ the quantities $$\underset{x\in \mathbb{R}}{\min }\sum _{\gamma \in \mathrm{\Gamma }}{e}^{-\pi \alpha (x-\gamma {)}^2}\text{and}\underset{x\in \mathbb{R}}{\max }\sum _{\gamma \in \mathrm{\Gamma }}{e}^{-\pi \alpha (x-\gamma {)}^2}$$ are maximized and minimized, respectively, if and only if the points are equispaced and whenever the number of points $n$ per period is sufficiently large (depending on $\alpha$). This solves the polarization problem for periodic configurations with a Gaussian weight on $\mathbb{R}$ for large $n$. The first result is shown using Fourier series. The second result follows from work of Cohn and Kumar on universal optimality and holds for all $n$ (independent of $\alpha$).