Maximal Polarization for Periodic Configurations on the Real Line

We prove that among all 1-periodic configurations $\Gamma $ of points on the real line $\mathbb{R}$ the quantities $\min _{x \in \mathbb{R}} \sum _{\gamma \in \Gamma } e<^>{- \pi \alpha (x - \gamma )<^>{2}}$ and $\max _{x \in \mathbb{R}} \sum _{\gamma \in \Gamma } e<^>{- \pi \alpha (x - \gamma )<^>{2}}$ 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 the work of Cohn and Kumar on universal optimality and holds for all $n$ (independent of $\alpha $).