Maximal polarization for periodic configurations on the real line
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866908336857808896 |
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| author | Faulhuber, Markus Steinerberger, Stefan |
| author_facet | Faulhuber, Markus Steinerberger, Stefan |
| contents | 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 $α$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_01532 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Maximal polarization for periodic configurations on the real line Faulhuber, Markus Steinerberger, Stefan Classical Analysis and ODEs 52C25, 74G65, 82B21 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 $α$). |
| title | Maximal polarization for periodic configurations on the real line |
| topic | Classical Analysis and ODEs 52C25, 74G65, 82B21 |
| url | https://arxiv.org/abs/2305.01532 |