Bounds on Discrete Potentials of Spherical (k,k)-Designs
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866908384565919744 |
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| author | Borodachov, S. Boyvalenkov, P. Saff, P. Dragnev. D. Hardin. E. Stoyanova, M. |
| author_facet | Borodachov, S. Boyvalenkov, P. Saff, P. Dragnev. D. Hardin. E. Stoyanova, M. |
| contents | We derive universal lower and upper bounds for max-min and min-max problems (also known as polarization) for the potential of spherical $(k,k)$-designs and provide certain examples, including unit-norm tight frames, that attain these bounds. The universality is understood in the sense that the bounds hold for all spherical $(k,k)$-designs and for a large class of potential functions, and the bounds involve certain nodes and weights that are independent of the potential. When the potential function is $h(t)=t^{2k}$, we prove an optimality property of the spherical $(k,k)$-designs in the class of all spherical codes of the same cardinality both for max-min and min-max potential problems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_00290 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds on Discrete Potentials of Spherical (k,k)-Designs Borodachov, S. Boyvalenkov, P. Saff, P. Dragnev. D. Hardin. E. Stoyanova, M. Metric Geometry Classical Analysis and ODEs Combinatorics 52C35, 05B30 We derive universal lower and upper bounds for max-min and min-max problems (also known as polarization) for the potential of spherical $(k,k)$-designs and provide certain examples, including unit-norm tight frames, that attain these bounds. The universality is understood in the sense that the bounds hold for all spherical $(k,k)$-designs and for a large class of potential functions, and the bounds involve certain nodes and weights that are independent of the potential. When the potential function is $h(t)=t^{2k}$, we prove an optimality property of the spherical $(k,k)$-designs in the class of all spherical codes of the same cardinality both for max-min and min-max potential problems. |
| title | Bounds on Discrete Potentials of Spherical (k,k)-Designs |
| topic | Metric Geometry Classical Analysis and ODEs Combinatorics 52C35, 05B30 |
| url | https://arxiv.org/abs/2411.00290 |