We prove that among all periodic configurations Γ of points on the
real line R the quantities \min_{x \in \mathbb{R}} \sum_{\gamma
\in \Gamma} e^{- \alpha (x - \gamma)^2} \quad \mbox{and} \quad \max_{x \in
\mathbb{R}} \sum_{\gamma \in \Gamma} e^{- \alpha (x - \gamma)^2} are
maximized and minimized, respectively, if and only if the points are equispaced
points whenever the number of points per period is sufficiently large
(depending on α). This solves the polarization problem for periodic
configurations with a Gaussian weight on R. The first result is
shown using Fourier series, the second follows from work of Cohn-Kumar on
universal optimality