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Constructive Approximation

, Volume 44, Issue 3, pp 385–415 | Cite as

Universal Lower Bounds for Potential Energy of Spherical Codes

  • P. G. Boyvalenkov
  • P. D. Dragnev
  • D. P. Hardin
  • E. B. Saff
  • M. M. Stoyanova
Article

Abstract

We derive and investigate lower bounds for the potential energy of finite spherical point sets (spherical codes). Our bounds are optimal in the following sense—they cannot be improved by employing polynomials of the same or lower degrees in the Delsarte–Yudin method. However, improvements are sometimes possible, and we provide a necessary and sufficient condition for the existence of such better bounds. All our bounds can be obtained in a unified manner that does not depend on the potential function, provided the potential is given by an absolutely monotone function of the inner product between pairs of points, and this is the reason we call them universal. We also establish a criterion for a given code of dimension n and cardinality N not to be LP-universally optimal; e.g., we show that two codes conjectured by Ballinger et al. to be universally optimal are not LP-universally optimal.

Keywords

Minimal energy problems Spherical potentials Spherical codes and designs Levenshtein bounds Delsarte–Goethals–Seidel bounds Linear programming 

Mathematics Subject Classification

74G65 94B65 (52A40, 05B30) 

Notes

Acknowledgments

We would like to thank Dr. Silvia Boumova for kindly allowing us to include the proof of Theorem 4.4(b) so as to make this article self-contained.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • P. G. Boyvalenkov
    • 1
    • 2
  • P. D. Dragnev
    • 3
  • D. P. Hardin
    • 4
  • E. B. Saff
    • 4
  • M. M. Stoyanova
    • 5
  1. 1.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria
  2. 2.Faculty of Mathematics and Natural SciencesSouth-Western UniversityBlagoevgradBulgaria
  3. 3.Department of Mathematical SciencesIndiana-Purdue UniversityFort WayneUSA
  4. 4.Center for Constructive Approximation, Department of MathematicsVanderbilt UniversityNashvilleUSA
  5. 5.Faculty of Mathematics and InformaticsSofia UniversitySofiaBulgaria

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