Advances in Computational Mathematics

, Volume 28, Issue 4, pp 331–354 | Cite as

The Coulomb energy of spherical designs on S2

Article

Abstract

In this work we give upper bounds for the Coulomb energy of a sequence of well separated spherical n-designs, where a spherical n-design is a set of m points on the unit sphere S2 ⊂ ℝ3 that gives an equal weight cubature rule (or equal weight numerical integration rule) on S2 which is exact for spherical polynomials of degree ⩽ n. (A sequence Ξ of m-point spherical n-designs X on S2 is said to be well separated if there exists a constant λ > 0 such that for each m-point spherical n-design X ∈ Ξ the minimum spherical distance between points is bounded from below by \(\frac{\lambda }{{{\sqrt m }}}\).) In particular, if the sequence of well separated spherical designs is such that m and n are related by m = O(n2), then the Coulomb energy of each m-point spherical n-design has an upper bound with the same first term and a second term of the same order as the bounds for the minimum energy of point sets on S2.

Keywords

acceleration of convergence Coulomb energy Coulomb potential equal weight cubature equal weight numerical integration orthogonal polynomials sphere spherical designs well separated point sets on sphere 

Mathematics Subject Classifications (2000)

Primary: 31C20 42C10 Secondary: 41A55 42C20 65B10 65D32 

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References

  1. 1.
    Anni, R., Connor, J.N.L., Noli, C.: Improved nearside-farside decomposition of elastic scattering amplitudes. Khimicheskaya Fizika 23(2), 6–12 (2004)Google Scholar
  2. 2.
    Antonov, V.A., Holševnikov, K.V.: An estimate of the remainder in the expansion of the generating function expansion for the Legendre polynomials (Generalization and improvement of the Bernstein’s inequality). Vestnik, Leningrad University. Mathematics 13, 163–166 (1981) (English translation)MATHGoogle Scholar
  3. 3.
    Bajnok, B., Damelin, S.B., Li, J., Mullen, G.L.: A constructive finite field method for scattering points on the surface of d-dimensional spheres. Computing 68(2), 97–109 (2002)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brauchart, J.S.: About the second term of the asymptotics for optimal Riesz energy on the sphere in the potential-theoretical case. Integral Transforms Spec. Funct. 17(5), 321–328 (2006)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chen, X., Womersley, R.S.: Existence of solutions to systems of underdetermined equations and spherical designs. SIAM J. Numer. Anal. 44(6), 2326–2341 (2006)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dahlberg, B.E.J.: On the distribution of Fekete points. Duke Math. J. 45(3), 537–542 (1978)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geom. Dedicata 6, 363–388 (1977)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Erber, T., Hockney, G.M.: Equilibrium configuration of N equal charges on a sphere. J. Phys. A. 24, L1369–L1377 (1991)CrossRefGoogle Scholar
  9. 9.
    Erdélyi, A. (ed.), Magnus, W., Oberhettinger, F., Tricomi, F.G. (research associates): Tables of Integral Transforms, vol. 2, Bateman Manuscript Project, California Institute of Technology. McGraw-Hill, New York, Toronto, London (1954)Google Scholar
  10. 10.
    Fejes Tóth, L.: On the densest packing of spherical caps. Amer. Math. Monthly 56(5), 330–331 (1949)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hardin, R.H., Sloane, N.J.A.: McLaren’s improved Snub cube and other new spherical designs in three dimensions. Discrete Comput. Geom. 15, 429–441 (1996)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hesse, K., Sloan, I.H.: Worst-case errors in a Sobolev space setting for cubature over the sphere S 2. Bull. Austral. Math. Soc. 71, 81–105 (2005)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Hesse, K., Sloan, I.H.: Cubature over the sphere S 2 in Sobolev spaces of arbitrary order. J. Approx. Theory 141(2), 118–133 (2006)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hobson, E.W.: On a general convergence theorem, and the theory of the representation of a function by series of normal functions. Proc. London Math. Soc. 6, 349–395 (1908)CrossRefGoogle Scholar
  15. 15.
    Hobson, E.W.: On the representation of a function by a series of Legendre’s functions. Proc. London Math. Soc. 7, 24–39 (1909)CrossRefGoogle Scholar
  16. 16.
    Korevaar, J., Meyers, J.L.H.: Spherical Faraday cage for the case of equal point charges and Chebyshev-type quadrature on the sphere. Integral Transform. Spec. Funct. 1(2), 105–117 (1993)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kuijlaars, A.B.J., Saff, E.B.: Asymptotics for minimal discrete energy on the sphere. Trans. Amer. Math. Soc. 350(2), 523–538 (1998)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lorch, L.: Alternative proof of a sharpened form of Bernstein’s inequality for Legendre polynomials. Appl. Anal. 14(3), 237–240 (1983)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lorch, L.: Alternative proof of a sharpened form of Bernstein’s inequality for Legendre polynomials: Corrigendum. Appl. Anal. 50, 47 (1993)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Rakhmanov, E.A., Saff, E.B., Zhou, Y.M.: Minimal discrete energy on the sphere. Math. Res. Lett. 1(6), 647–662 (1994)MATHMathSciNetGoogle Scholar
  21. 21.
    Rakhmanov, E.A., Saff, E.B., Zhou, Y.M.: Electrons on the sphere. In: Computational Methods and Function Theory 1994 (Penang), Series in Approximations and Decompositions, No 5, pp. 293–309. World Scientific Publishing, River Edge, NJ (1995)Google Scholar
  22. 22.
    Reimer, M.: Hyperinterpolation on the sphere at the minimal projection order. J. Approx. Theory 104, 272–286 (2000)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Sansone, G.: Orthogonal Functions. Interscience, New York (1959)MATHGoogle Scholar
  24. 24.
    Szegö, G.: Orthogonal polynomials. In: American Mathematical Society Colloquium Publications, 4th edn, vol. 23. American Mathematical Society, Providence, RI (1975)Google Scholar
  25. 25.
    Wagner, G.: On means of distances on the surface of a sphere (lower bounds). Pacific J. Math. 144(2), 389–398 (1990)MATHMathSciNetGoogle Scholar
  26. 26.
    Wagner, G.: On means of distances on the surface of a sphere. II (upper bounds). Pacific J. Math. 154(2), 381–396 (1992)MATHMathSciNetGoogle Scholar
  27. 27.
    Yennie, D.R., Ravenhall, D.G., Wilson, R.N.: Phase-shift calculation of high-energy electron scattering. Phys. Rev. 95(2), 500–512 (July 1954)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsThe University of New South WalesSydneyAustralia

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