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We prove that among all 1-periodic configurations $\Gamma$ of points on thereal line $\mathbb{R}$ the quantities $$ \min_{x \in \mathbb{R}} \sum_{\gamma\in \Gamma} e^{- \pi \alpha (x - \gamma)^2} \quad \text{and} \quad \max_{x \in\mathbb{R}} \sum_{\gamma \in \Gamma} e^{- \pi \alpha (x - \gamma)^2}$$ aremaximized and minimized, respectively, if and only if the points are equispacedand whenever the number of points $n$ per period is sufficiently large(depending on $\alpha$). This solves the polarization problem for periodicconfigurations with a Gaussian weight on $\mathbb{R}$ for large $n$. The firstresult is shown using Fourier series. The second result follows from work ofCohn and Kumar on universal optimality and holds for all $n$ (independent of$\alpha$).

Maximal polarization for periodic configurations on the real line