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- Title
Maximal Polarization for Periodic Configurations on the Real Line.
- Authors
Faulhuber, Markus; Steinerberger, Stefan
- Abstract
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 $|).
- Subjects
FOURIER series; PROBLEM solving
- Publication
IMRN: International Mathematics Research Notices, 2024, Vol 2024, Issue 9, p7914
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnae003