Advances in Computational Mathematics

, Volume 39, Issue 1, pp 27–43 | Cite as

Discrepancy, separation and Riesz energy of finite point sets on the unit sphere

Article

Abstract

For \(d \geqslant 2,\) we consider asymptotically equidistributed sequences of \(\mathbb S^d\) codes, with an upper bound \(\operatorname{\boldsymbol{\delta}}\) on spherical cap discrepancy, and a lower bound Δ on separation. For such sequences, if 0 < s < d, then the difference between the normalized Riesz s energy of each code, and the normalized s-energy double integral on the sphere is bounded above by \(\operatorname{O}\big(\operatorname{\boldsymbol{\delta}}^{1-s/d}\,\Delta^{-s}\,N^{-s/d}\big),\) where N is the number of code points. For well separated sequences of spherical codes, this bound becomes \(\operatorname{O}\big(\operatorname{\boldsymbol{\delta}}^{1-s/d}\big).\) We apply these bounds to minimum energy sequences, sequences of well separated spherical designs, sequences of extremal fundamental systems, and sequences of equal area points.

Keywords

Sphere Spherical cap discrepancy Separation Riesz energy 

Mathematics Subject Classifications (2010)

11K38 41A55 65D30 

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References

  1. 1.
    Beck, J.: New results in the theory of irregularities of point distributions. In: Number Theory Noordwijkerhout 1983. Lecture Notes in Mathematics, vol. 1068, pp. 1–16. Springer, Berlin (1984)CrossRefGoogle Scholar
  2. 2.
    Beck, J., Chen, W.: Irregularities of Distribution. Cambridge University Press (1987)Google Scholar
  3. 3.
    Blümlinger, M.: Asymptotic distribution and weak convergence on compact Riemannian manifolds. Monatshefte für Mathematik 110, 177–188 (1990). doi: 10.1007/BF01301674 MATHCrossRefGoogle Scholar
  4. 4.
    Bondarenko, A., Radchenko, D., Viazovska, M.: Optimal asymptotic bounds for spherical designs (2011). ArXiv:1009.4407v3 [math.MG]
  5. 5.
    Brauchart, J.S.: Invariance principles for energy functionals on spheres. Mon.hefte Math. 141(2), 101–117 (2004)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Brauchart, J.S.: Points on an Unit Sphere in R d + 1, Riesz Energy, Discrepancy and Numerical Integration. PhD thesis, Institut für Mathematik A, Technische Universität Graz. Graz, Austria (2005)Google Scholar
  7. 7.
    Brauchart, J.S.: Optimal logarithmic energy points on the unit sphere. Math. Comput. 77(263), 1599–1613 (2008)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Brauchart, J.S., Dick, J.: A simple proof of Stolarsky’s invariance principle. In: Proceedings of the American Mathematical Society (2011, in press). ArXiv:1101.4448v1 [math.NA]
  9. 9.
    Damelin, S.B.: A walk through energy, discrepancy, numerical integration and group invariant measures on measurable subsets of Euclidean space. Numer. Algorithms 48(1–3), 213–235 (2008)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Damelin, S.B., Grabner, P.J.: Energy functionals, numerical integration and asymptotic equidistribution on the sphere. J. Complex. 19(3), 231–246 (2003); (Postscript) Corrigendum. J. Complex. 20, 883–884 (2004)Google Scholar
  11. 11.
    Damelin, S.B., Grabner, P.J.: Corrigendum to Energy functionals, numerical integration and asymptotic equidistribution on the sphere. J. Complex 19, 231–246 (2003); J. Complex. 20, 883–884 (2004)Google Scholar
  12. 12.
    Damelin, S.B., Hickernell, F.J., Ragozin, D.L., Zeng, X.: On energy, discrepancy and group invariant measures on measurable subsets of Euclidean space. J. Fourier Anal. Appl. 16(6), 813–839 (2010)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Damelin, S.B., Maymeskul, V.: On point energies, separation radius and mesh norm for s-extremal configurations on compact sets in ℝn. J. Complex. 21(6), 845–863 (2005)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geom. Dedic. 6, 363–388 (1977)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Dragnev, P.D., Saff, E.B.: Riesz spherical potentials with external fields and minimal energy points separation. Potential Anal. 26(2), 139–162 (2007)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Götz, M.: On the distribution of weighted extremal points on a surface in ℝd, d ⩾ 3. Potential Anal. 13, 345–359 (2000)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Götz, M.: On the Riesz energy of measures. J. Approx. Theory 122(1), 62–78 (2003)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Grabner, P.J.: Erdös-Turán type discrepancy bounds. Mon.hefte Math. 111(2), 127–135 (1991)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Grabner, P.J., Tichy, R.F.: Spherical designs, discrepancy and numerical integration. Math. Comput. 60, 327–336 (1993)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Hardin, D.P., Saff, E.B.: Minimal Riesz energy point configurations for rectifiable d-dimensional manifolds. Adv. Math. 193(1), 174–204 (2005)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Hesse, K.: The s-energy of spherical designs on S 2. Adv. Comput. Math. 30(1), 37–59 (2009)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Hesse, K., Leopardi, P.: The Coulomb energy of spherical designs on S 2. Adv. Comput. Math. 28(4), 331–354 (2008). doi: 10.1007/s10444-007-9026-7 MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Korevaar, J., Meyers, J.L.H.: Spherical Faraday cage for the case of equal point charges and Chebyshev-type quadrature on the sphere. Integral Transforms Spec. Funct. 1(2), 105–117 (1993)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Kuijlaars, A.B.J., Saff, E.B.: Asymptotics for minimal discrete energy on the sphere. Trans. Am. Math. Soc. 350(2), 523–538 (1998)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Kuijlaars, A.B.J., Saff, E.B., Sun, X.: On separation of minimal Riesz nergy points on spheres in Euclidean spaces. J. Comput. Appl. Math. 199(1), 172–180 (2007)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Landkof, N.S.: Foundations of Modern Potential Theory. Springer, Berlin (1972). Translated from the Russian by A.P. DoohovskoyGoogle Scholar
  27. 27.
    Leopardi, P.: Distributing Points on the Sphere: Partitions, Separation, Quadrature and Energy. Ph.D. thesis, The University of New South Wales (2007)Google Scholar
  28. 28.
    Levesley, J., Sun, X.: Approximating probability measures on manifolds via radial basis functions. In: Georgoulis, E.H., Iske, A., Levesley, J. (eds.) Approximation Algorithms for Complex Systems, Springer Proceedings in Mathematics, vol. 3, pp. 151–180. Springer, Berlin (2011)CrossRefGoogle Scholar
  29. 29.
    Lubotzky, A., Phillips, R., Sarnak, P.: Hecke operators and distributing points on S 2 I. Commun. Pure Appl. Math. 39, S149–S186 (1986)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Marzo, J., Ortega-Cerdà, J.: Equidistribution of Fekete points on the sphere. Constr. Approx. 32(3), 5139–521 (2010). doi: 10.1007/s00365-009-9051-5 CrossRefGoogle Scholar
  31. 31.
    Müller, C.: Spherical Harmonics. Lecture Notes in Mathematics, vol. 17. Springer, Berlin (1966)MATHGoogle Scholar
  32. 32.
    Rakhmanov, E.A., Saff, E.B., Zhou, Y.M.: Electrons on the sphere. In: Computational Methods and Function Theory 1994 (Penang), no. 5 in Series in Approximations and Decompositions, pp. 293–309. World Scientific Publishing, River Edge (1995)Google Scholar
  33. 33.
    Rankin, R.A.: The closest packing of spherical caps in n dimensions. Proc. Glasgow Math. Assoc 2, 139–144 (1955)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Rao, R.R.: Relations between weak and uniform convergence of measures with applications. Ann. Math. Stat. 33, 659–680 (1962)MATHCrossRefGoogle Scholar
  35. 35.
    Reimer, M.: Constructive Theory of Multivariate Functions. BI Wissenschaftsverlag, Mannheim, Wien, Zürich (1990)Google Scholar
  36. 36.
    Sloan, I.H., Womersley, R.S.: Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math. 21, 107–125 (2004)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Stolarsky, K.B.: Sums of distances between points on a sphere II. Proc. Am. Math. Soc. 41, 575–582 (1973)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Tammes, P.M.L.: On the origin of number and arrangements of the places of exit on the surface of pollen-grains. Recl. Trav. Bot. Néerl. 27, 1–84 (1930)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Centre for Mathematics and its ApplicationsAustralian National UniversityCanberraAustralia

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