Maximal polarization for periodic configurations on the real line

TL;DR

This work solves the 1D Gaussian polarization problem for periodic configurations in the regime of many points per period. Using Fourier analysis, Poisson summation, and theta-function technology, it proves that equispaced configurations maximize the minimum of the Gaussian-summed field $p_α(x)$ and, via a universal optimality argument, minimize its maximum, with the results holding for large $n$ (depending only on $α$). The analysis hinges on a delicate perturbative regime showing any near-equispaced configuration yields a strictly smaller minimum unless the configuration is exactly equispaced; a complementary corollary to heat-equation sampling extends the result to sampling on the circle. The findings connect to energy minimization and universal optimality literature, and they establish a robust Fourier-analytic approach applicable to a broader class of fast-decaying kernels beyond the Gaussian.

Abstract

We prove that among all 1-periodic configurations $Γ$ of points on the real line $\mathbb{R}$ the quantities $$ \min_{x \in \mathbb{R}} \sum_{γ\in Γ} e^{- πα(x - γ)^2} \quad \text{and} \quad \max_{x \in \mathbb{R}} \sum_{γ\in Γ} e^{- πα(x - γ)^2}$$ 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 $α$). 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 $α$).

Figures (2)

• Figure 1: Illustration of the result of Cohn and Kumar CohKum07. Building the average of $p_\alpha(x)$ at the points $\{x_1, \ldots , x_n\}$ (in this case $n=3$) for periodic, non-equispaced configuration always yields a larger value than for the equispaced points. As we sum $n$ times the maximum in the equispaced case, it follows that the maximum of $p_\alpha(x)$ is minimal only for the equispaced configuration.
• Figure 2: For the sum of equispaced periodic Gaussians the minimum is achieved midway between successive shifts. For sums of shifts by a general periodic configurations it is rather difficult to grasp the minimum. For the plot we have normalized the sum to oscillate around 1, i.e., the integral over 1 period is 1.

Theorems & Definitions (26)

• Proposition 1.1: Application of Cohn and Kumar CohKum07
• Theorem : Main Result
• Corollary 2.4.1
• proof
• Remark
• proof
• Proposition 5.1.1
• proof
• Lemma 5.2.1
• proof