Energy, Polarization, and Separation of Greedy Sequences for Riesz and Green Kernels
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866918449688608768 |
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| author | Bilyk, Dmitriy Kryvonos, Liudmyla Matzke, Ryan W. Saff, Edward |
| author_facet | Bilyk, Dmitriy Kryvonos, Liudmyla Matzke, Ryan W. Saff, Edward |
| contents | We investigate the asymptotic behavior of greedy $s$-Riesz and Green energy sequences $\{x_{n}\}_{n=1}^{\infty}$ on the unit sphere $\mathbb{S}^{d} \subset \mathbb{R}^{d+1}$, where each point $x_n$ is defined as the minimizer of the discrete potential generated by the preceding points $x_1, x_2, ..., x_{n-1}$. We show that the greedy sequence attains optimal growth behavior for the second-order term of the Green and Riesz $s$-energies when $d-2 \leq s < d$. The main idea is to establish the bounds on polarization using well-separation properties of the greedy configurations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14326 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Energy, Polarization, and Separation of Greedy Sequences for Riesz and Green Kernels Bilyk, Dmitriy Kryvonos, Liudmyla Matzke, Ryan W. Saff, Edward Classical Analysis and ODEs 31C20, 31B05 We investigate the asymptotic behavior of greedy $s$-Riesz and Green energy sequences $\{x_{n}\}_{n=1}^{\infty}$ on the unit sphere $\mathbb{S}^{d} \subset \mathbb{R}^{d+1}$, where each point $x_n$ is defined as the minimizer of the discrete potential generated by the preceding points $x_1, x_2, ..., x_{n-1}$. We show that the greedy sequence attains optimal growth behavior for the second-order term of the Green and Riesz $s$-energies when $d-2 \leq s < d$. The main idea is to establish the bounds on polarization using well-separation properties of the greedy configurations. |
| title | Energy, Polarization, and Separation of Greedy Sequences for Riesz and Green Kernels |
| topic | Classical Analysis and ODEs 31C20, 31B05 |
| url | https://arxiv.org/abs/2604.14326 |