Bounds on energy and potentials of discrete measures on the sphere
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909423206662144 |
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| author | Borodachov, S. Boyvalenkov, P. Dragnev, P. Hardin, D. Saff, E. Stoyanova, M. |
| author_facet | Borodachov, S. Boyvalenkov, P. Dragnev, P. Hardin, D. Saff, E. Stoyanova, M. |
| contents | We establish upper and lower universal bounds for potentials of weighted designs on the sphere $\mathbb{S}^{n-1}$ that depend only on quadrature nodes and weights derived from the design structure. Our bounds hold for a large class of potentials that includes absolutely monotone functions. The classes of spherical designs attaining these bounds are characterized. Additionally, we study the problem of constrained energy minimization for Borel probability measures on $\mathbb{S}^{n-1}$ and apply it to optimal distribution of charge supported at a given number of points on the sphere. In particular, our results apply to $p$-frame energy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07442 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds on energy and potentials of discrete measures on the sphere Borodachov, S. Boyvalenkov, P. Dragnev, P. Hardin, D. Saff, E. Stoyanova, M. Metric Geometry Classical Analysis and ODEs Probability We establish upper and lower universal bounds for potentials of weighted designs on the sphere $\mathbb{S}^{n-1}$ that depend only on quadrature nodes and weights derived from the design structure. Our bounds hold for a large class of potentials that includes absolutely monotone functions. The classes of spherical designs attaining these bounds are characterized. Additionally, we study the problem of constrained energy minimization for Borel probability measures on $\mathbb{S}^{n-1}$ and apply it to optimal distribution of charge supported at a given number of points on the sphere. In particular, our results apply to $p$-frame energy. |
| title | Bounds on energy and potentials of discrete measures on the sphere |
| topic | Metric Geometry Classical Analysis and ODEs Probability |
| url | https://arxiv.org/abs/2412.07442 |