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.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1972)
Atkinson, K.: An Introduction to Numerical Analysis, 2nd edn. John Wiley & Sons, New York (1989)
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)
Bernstein, S.N.: On quadrature formulas with positive coefficients. Izv. Akad. Nauk. SSSR, Ser. Mat. 1(4), 479–503 (1937, Russian)
Bernstein, S.N.: Sur un système d’équations indéterminées. J. Math. Pures Appl. 17(9), 179–186 (1938)
Bondarenko, A.V., Viazovska, M.S.: New asymptotic estimates for spherical designs. J. Approx. Theory 152, 101–106 (2008)
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)
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)
Dai, F., Wang, H.: Positive cubature formulas and Marcinkiewicz–Zygmund inequalities on spherical caps. Constr. Approximation 31, 1–36 (2010)
Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geom. Dedic. 6, 363–388 (1977)
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)
Freeden, W.: Multiscale Modelling of Spaceborne Geodata. B.G. Teubner, Stuttgart, Leipzig (1999)
Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere with Applications to Geomathematics. Oxford University Press, Oxford (1998)
Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Oxford University Press, New York (2004)
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)
Hesse, K.: Complexity of numerical integration over spherical caps in a Sobolev space setting. J. Complex. 27, 383–403 (2011)
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)
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)
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)
Kuijlaars, A.: The minimal number of nodes in Chebyshev-type quadrature formulas. Indag. Math. N. S., 4(3), 339–362 (1993)
Markov, A.A.: On a problem of D.I. Mendeleev. In: Selected Works (in Russian), GITTL, Moscow-Leningrad (1948)
Mhaskar, H.N.: Local quadrature formulas on the sphere. J. Complex. 20, 753–772 (2004)
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)
Mhaskar, H.N.: Weighted quadrature formulas and approximation by zonal function networks on the sphere. J. Complex. 22, 348–370 (2006)
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))
Petrushev, P.P.: Approximation by ridge functions and neural networks. SIAM J. Math. Anal. 30, 155–189 (1998)
Reimer, M.: Hyperinterpolation on the sphere at the minimal projection order. J. Approx. Theory 104, 272–286 (2000)
Reimer, M.: Multivariate Polynomial Approximation. Birkhäuser Verlag, Basel, Boston, Berlin (2003)
Sansone, G.: Orthogonal Functions. Interscience Publishers, London, New York (1959)
Sloan, I.H., Womersley, R.S.: Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math. 21, 107–125 (2004)
Sloan, I.H., Womersley, R.S.: A variational characterization of spherical designs. J. Approx. Theory 159, 308–318 (2009)
Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall Inc., Englewood Cliffs, NJ (1971)
Szegö, G.: Orthogonal polynomials. In: American Mathematical Society Colloquium Publications, 4th edn., vol. 23. American Mathematical Society, Providence, Rhode Island (1975)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by I. H. Sloan.
Rights and permissions
About this article
Cite this article
Hesse, K., Womersley, R.S. Numerical integration with polynomial exactness over a spherical cap. Adv Comput Math 36, 451–483 (2012). https://doi.org/10.1007/s10444-011-9187-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-011-9187-2
Keywords
- Equal weight rules
- Markov inequalities
- Numerical integration
- Polynomial exactness
- Positive weights
- Spherical cap
- Tensor product rules