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