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Advances in Computational Mathematics

, Volume 36, Issue 3, pp 451–483 | Cite as

Numerical integration with polynomial exactness over a spherical cap

  • Kerstin HesseEmail author
  • Robert S. Womersley
Article

Abstract

This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere \(\mathbb{S}^2\), we discuss tensor product rules with n 2/2 + O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree ≤ n, and can be easily and inexpensively implemented. Numerical tests illustrate the performance of these rules. A similar derivation establishes the existence of equal weight rules with degree of polynomial exactness n and O(n 3) nodes for numerical integration over spherical caps on \(\mathbb{S}^2\). For arbitrary d ≥ 2, this strategy is extended to provide rules for numerical integration over spherical caps on \(\mathbb{S}^d\) that have O(n d ) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree ≤ n. We also show that positive weight rules for numerical integration over spherical caps on \(\mathbb{S}^d\) that are exact for all spherical polynomials of degree ≤ n have at least O(n d ) nodes and possess a certain regularity property.

Keywords

Equal weight rules Markov inequalities Numerical integration Polynomial exactness Positive weights Spherical cap Tensor product rules 

Mathematics Subject Classifications (2010)

Primary 65D30 65D32 33C55; Secondary 41A55 33C50 42C10 43A90 58C35 

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References

  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1972)Google Scholar
  2. 2.
    Atkinson, K.: An Introduction to Numerical Analysis, 2nd edn. John Wiley & Sons, New York (1989)zbMATHGoogle Scholar
  3. 3.
    Bernstein, S.N.: Sur les formules de quadrature de Cotes et Tchebycheff. C. R. Acad. Sci. URSS (Dokl. Akad. Nauk SSSR), N. S. 14, 323–327 (1937)Google Scholar
  4. 4.
    Bernstein, S.N.: On quadrature formulas with positive coefficients. Izv. Akad. Nauk. SSSR, Ser. Mat. 1(4), 479–503 (1937, Russian)Google Scholar
  5. 5.
    Bernstein, S.N.: Sur un système d’équations indéterminées. J. Math. Pures Appl. 17(9), 179–186 (1938)zbMATHGoogle Scholar
  6. 6.
    Bondarenko, A.V., Viazovska, M.S.: New asymptotic estimates for spherical designs. J. Approx. Theory 152, 101–106 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Brown, G., Dai, F., Sun, Y.S.: Kolmogorov widths of classes of smooth functions on the sphere S d − 1. J. Complex. 18, 1001–1023 (2002)zbMATHCrossRefGoogle Scholar
  8. 8.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dai, F., Wang, H.: Positive cubature formulas and Marcinkiewicz–Zygmund inequalities on spherical caps. Constr. Approximation 31, 1–36 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geom. Dedic. 6, 363–388 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Erdélyi, A. (ed.), Magnus, W., Oberhettinger, F., Tricomi, F.G. (research associates): Higher Transcendental Functions, vol. 2. Bateman Manuscript Project, California Institute of Technology. McGraw-Hill, New York, Toronto, London (1953)Google Scholar
  12. 12.
    Freeden, W.: Multiscale Modelling of Spaceborne Geodata. B.G. Teubner, Stuttgart, Leipzig (1999)Google Scholar
  13. 13.
    Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere with Applications to Geomathematics. Oxford University Press, Oxford (1998)zbMATHGoogle Scholar
  14. 14.
    Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Oxford University Press, New York (2004)zbMATHGoogle Scholar
  15. 15.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Hesse, K.: Complexity of numerical integration over spherical caps in a Sobolev space setting. J. Complex. 27, 383–403 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Hesse, K., Sloan, I.H.: Optimal order integration on the sphere. In: Li, T., Zhang, P. (eds.) Frontiers and Prospects of Contemporary Applied Mathematics, Series in Contemporary Applied Mathematics CAM 6, pp. 59–70. Higher Education Press and World Scientific (2005)Google Scholar
  18. 18.
    Korevaar, J.: Chebyshev-type quadratures; use of complex analysis and potential theory. In: Gauthier, P.M. (ed.), Sabidussi, G. (techn. ed.) Complex Potential Theory, pp. 325–364. Kluwer Academic Publishers, Dordrecht, Boston (1994)Google Scholar
  19. 19.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Kuijlaars, A.: The minimal number of nodes in Chebyshev-type quadrature formulas. Indag. Math. N. S., 4(3), 339–362 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Markov, A.A.: On a problem of D.I. Mendeleev. In: Selected Works (in Russian), GITTL, Moscow-Leningrad (1948)Google Scholar
  22. 22.
    Mhaskar, H.N.: Local quadrature formulas on the sphere. J. Complex. 20, 753–772 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Mhaskar, H.N.: Local quadrature formulas on the sphere, II. In: Neamtu, M., Saff, E.B. (eds.) Advances in Constructive Approximation, pp. 333–344. Nashboro Press, Nashville (2004)Google Scholar
  24. 24.
    Mhaskar, H.N.: Weighted quadrature formulas and approximation by zonal function networks on the sphere. J. Complex. 22, 348–370 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Mhaskar, H.N., Narcowich, F.J., Ward, J.D.: Spherical Marcinkiewicz–Zygmund inequalities and positive quadrature. Math. Comput. 70, 1113–1130 (2001) (Corrigendum: Math. Comput. 71, 453–454 (2002))MathSciNetzbMATHGoogle Scholar
  26. 26.
    Petrushev, P.P.: Approximation by ridge functions and neural networks. SIAM J. Math. Anal. 30, 155–189 (1998)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Reimer, M.: Hyperinterpolation on the sphere at the minimal projection order. J. Approx. Theory 104, 272–286 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Reimer, M.: Multivariate Polynomial Approximation. Birkhäuser Verlag, Basel, Boston, Berlin (2003)zbMATHCrossRefGoogle Scholar
  29. 29.
    Sansone, G.: Orthogonal Functions. Interscience Publishers, London, New York (1959)zbMATHGoogle Scholar
  30. 30.
    Sloan, I.H., Womersley, R.S.: Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math. 21, 107–125 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Sloan, I.H., Womersley, R.S.: A variational characterization of spherical designs. J. Approx. Theory 159, 308–318 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall Inc., Englewood Cliffs, NJ (1971)zbMATHGoogle Scholar
  33. 33.
    Szegö, G.: Orthogonal polynomials. In: American Mathematical Society Colloquium Publications, 4th edn., vol. 23. American Mathematical Society, Providence, Rhode Island (1975)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Handelshochschule Leipzig gGmbHLeipzigGermany
  2. 2.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

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