Skip to main content

Advertisement

SpringerLink
Log in

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

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Chapter  Google Scholar 

  2. Beck, J., Chen, W.: Irregularities of Distribution. Cambridge University Press (1987)

  3. Blümlinger, M.: Asymptotic distribution and weak convergence on compact Riemannian manifolds. Monatshefte für Mathematik 110, 177–188 (1990). doi:10.1007/BF01301674

    Article  MATH  Google Scholar 

  4. Bondarenko, A., Radchenko, D., Viazovska, M.: Optimal asymptotic bounds for spherical designs (2011). ArXiv:1009.4407v3 [math.MG]

  5. Brauchart, J.S.: Invariance principles for energy functionals on spheres. Mon.hefte Math. 141(2), 101–117 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  7. Brauchart, J.S.: Optimal logarithmic energy points on the unit sphere. Math. Comput. 77(263), 1599–1613 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geom. Dedic. 6, 363–388 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Götz, M.: On the distribution of weighted extremal points on a surface in ℝd, d ⩾ 3. Potential Anal. 13, 345–359 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Götz, M.: On the Riesz energy of measures. J. Approx. Theory 122(1), 62–78 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Grabner, P.J.: Erdös-Turán type discrepancy bounds. Mon.hefte Math. 111(2), 127–135 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  19. Grabner, P.J., Tichy, R.F.: Spherical designs, discrepancy and numerical integration. Math. Comput. 60, 327–336 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hardin, D.P., Saff, E.B.: Minimal Riesz energy point configurations for rectifiable d-dimensional manifolds. Adv. Math. 193(1), 174–204 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hesse, K.: The s-energy of spherical designs on S 2. Adv. Comput. Math. 30(1), 37–59 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Landkof, N.S.: Foundations of Modern Potential Theory. Springer, Berlin (1972). Translated from the Russian by A.P. Doohovskoy

  27. Leopardi, P.: Distributing Points on the Sphere: Partitions, Separation, Quadrature and Energy. Ph.D. thesis, The University of New South Wales (2007)

  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)

    Chapter  Google Scholar 

  29. Lubotzky, A., Phillips, R., Sarnak, P.: Hecke operators and distributing points on S 2 I. Commun. Pure Appl. Math. 39, S149–S186 (1986)

    Article  MathSciNet  Google Scholar 

  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

    Article  Google Scholar 

  31. Müller, C.: Spherical Harmonics. Lecture Notes in Mathematics, vol. 17. Springer, Berlin (1966)

    MATH  Google Scholar 

  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. Rankin, R.A.: The closest packing of spherical caps in n dimensions. Proc. Glasgow Math. Assoc 2, 139–144 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rao, R.R.: Relations between weak and uniform convergence of measures with applications. Ann. Math. Stat. 33, 659–680 (1962)

    Article  MATH  Google Scholar 

  35. Reimer, M.: Constructive Theory of Multivariate Functions. BI Wissenschaftsverlag, Mannheim, Wien, Zürich (1990)

  36. Sloan, I.H., Womersley, R.S.: Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math. 21, 107–125 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  37. Stolarsky, K.B.: Sums of distances between points on a sphere II. Proc. Am. Math. Soc. 41, 575–582 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  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 

Download references

Author information

Authors and Affiliations

  1. Centre for Mathematics and its Applications, Australian National University, Canberra, 0200, Australia

    Paul Leopardi

Authors
  1. Paul Leopardi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul Leopardi.

Additional information

Communicated by Yang Wang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Leopardi, P. Discrepancy, separation and Riesz energy of finite point sets on the unit sphere. Adv Comput Math 39, 27–43 (2013). https://doi.org/10.1007/s10444-011-9266-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-011-9266-4

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation