Explicit Families of Functions on the Sphere with Exactly Known Sobolev Space Smoothness

  • Johann S. Brauchart


We analyze explicit trial functions defined on the unit sphere \({\mathbb {S}}^d\) in the Euclidean space \({\mathbb {R}}^{d+1}\), d ≥ 1, that are integrable in the \({\mathbb {L}}_p\)-sense, p ∈ [1, ). These functions depend on two free parameters: one determines the support and one, a critical exponent, controls the behavior near the boundary of the support. Three noteworthy features are: (1) they are simple to implement and capture typical behavior of functions in applications, (2) their integrals with respect to the uniform measure on the sphere are given by explicit formulas and, thus, their numerical values can be computed to arbitrary precision, and (3) their smoothness can be defined a priori, that is to say, they belong to Sobolev spaces \({\mathbb {H}}^s({\mathbb {S}}^d)\) up to a specified index \(\bar {s}\) determined by the parameters of the function. Considered are zonal functions g(x) = h(x ⋅p), where p is some fixed pole on \({\mathbb {S}}^d\). The function h(t) is of the type \([ \max \{t,T\} ]^\alpha \) or a variation of a truncated power function \(x \mapsto (x)_+^\alpha \) (which assumes 0 if x ≤ 0 and is the power xα if x > 0) that reduces to \([ \max \{t-T,0\} ]^\alpha \), \([ \max \{t^2-T^2,0\} ]^{\alpha }\), and \([ \max \{T^2-t^2,0\} ]^{\alpha }\) if α > 0. These types of trial functions have as support the whole sphere, a spherical cap centered at p, a bi-cap centered at the antipodes p, −p, or an equatorial belt. We give inclusion theorems that identify the critical smoothness \(\bar {s} = \bar {s}(T,\alpha )\) and explicit formulas for the integral over the sphere. We obtain explicit formulas for the coefficients in the Laplace-Fourier expansion of these trial functions and provide the leading order term in the asymptotics for large index of the coefficients.



The research of this author was supported, in part, by the Austrian Science Fund FWF project F5510 (part of the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory and Applications”) and M2030 Lise Meitner Programm “Self organization by local interaction”.


  1. 1.
    An, C., Chen, X., Sloan, I.H., Womersley, R.S.: Well conditioned spherical designs for integration and interpolation on the two-sphere. SIAM J. Numer. Anal. 48(6), 2135–2157 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    An, C., Chen, X., Sloan, I.H., Womersley, R.S.: Regularized least squares approximations on the sphere using spherical designs. SIAM J. Numer. Anal. 50(3), 1513–1534 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Berens, H., Butzer, P.L., Pawelke, S.: Limitierungsverfahren von Reihen mehrdimensionaler Kugelfunktionen und deren Saturationsverhalten. Publ. Res. Inst. Math. Sci. Ser. A 4, 201–268 (1968/1969)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bilyk, D., Lacey, T.: One bit sensing, discrepancy, and Stolarsky principle. Mat. Sb. 208(6), 4–25 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bondarenko, A., Radchenko, D., Viazovska, M.: Optimal asymptotic bounds for spherical designs. Ann. Math. (2) 178(2), 443–452 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Brauchart, J.S., Dick, J.: Quasi-Monte Carlo rules for numerical integration over the unit sphere \(\mathbb S^2\). Numer. Math. 121(3), 473–502 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Brauchart, J.S., Dick, J.: A characterization of Sobolev spaces on the sphere and an extension of Stolarsky’s invariance principle to arbitrary smoothness. Constr. Approx. 38(3), 397–445 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Brauchart, J.S., Dick, J.: A simple proof of Stolarsky’s invariance principle. Proc. Am. Math. Soc. 141(6), 2085–2096 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Brauchart, J.S., Hesse, K.: Numerical integration over spheres of arbitrary dimension. Constr. Approx. 25(1), 41–71 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Brauchart, J.S., Saff, E.B., Sloan, I.H., Womersley, R.S.: QMC designs: optimal order quasi Monte Carlo integration schemes on the sphere. Math. Comput. 83(290), 2821–2851 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Brauchart, J.S., Dick, J., Saff, E.B., Sloan, I.H., Wang, Y.G., Womersley, R.S.: Covering of spheres by spherical caps and worst-case error for equal weight cubature in Sobolev spaces. J. Math. Anal. Appl. 431(2), 782–811 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Brauchart, J.S., Dick, J., Fang, L.: Spatial low-discrepancy sequences, spherical cone discrepancy, and applications in financial modeling. J. Comput. Appl. Math. 286, 28–53 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Chen, X., Womersley, R.S.: Existence of solutions to systems of underdetermined equations and spherical designs. SIAM J. Numer. Anal. 44(6), 2326–2341 (electronic) (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Chen, X., Frommer, A., Lang, B.: Computational existence proofs for spherical t-designs. Numer. Math. 117(2), 289–305 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geom. Dedicata 6(3), 363–388 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Frenzen, C.L., Wong, R.: A uniform asymptotic expansion of the Jacobi polynomials with error bounds. Can. J. Math. 37(5), 979–1007 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Genz, A.: Testing multidimensional integration routines. In: Proceedings of International Conference on Tools, Methods and Languages for Scientific and Engineering Computation, pp. 81–94. Elsevier North-Holland, Inc., New York, NY (1984)Google Scholar
  18. 18.
    Genz, A.: Fully symmetric interpolatory rules for multiple integrals over hyper-spherical surfaces. J. Comput. Appl. Math. 157(1), 187–195 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Grabner, P.J., Klinger, B., Tichy, R.F.: Discrepancies of point sequences on the sphere and numerical integration. In: Multivariate Approximation (Witten-Bommerholz, 1996). Mathematical Research, vol. 101, pp. 95–112. Akademie Verlag, Berlin (1997)Google Scholar
  20. 20.
    Hahn, E.: Asymptotik bei Jacobi-Polynomen und Jacobi-Funktionen. Math. Z. 171(3), 201–226 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Hesse, K.: A lower bound for the worst-case cubature error on spheres of arbitrary dimension. Numer. Math. 103(3), 413–433 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Hesse, K., Sloan, I.H.: Optimal lower bounds for cubature error on the sphere S2. J. Complexity 21(6), 790–803 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Hesse, K., Sloan, I.H.: Worst-case errors in a Sobolev space setting for cubature over the sphere S2. Bull. Aust. Math. Soc. 71(1), 81–105 (2005)zbMATHCrossRefGoogle Scholar
  24. 24.
    Hesse, K., Sloan, I.H.: Cubature over the sphere S2 in Sobolev spaces of arbitrary order. J. Approx. Theory 141(2), 118–133 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Hesse, K., Sloan, I.H.: Hyperinterpolation on the sphere. In: Frontiers in Interpolation and Approximation. Pure and Applied Mathematics (Boca Raton), vol. 282, pp. 213–248. Chapman & Hall/CRC, Boca Raton, FL (2007)CrossRefGoogle Scholar
  26. 26.
    Hesse, K., Sloan, I.H., Womersley, R.S.: Numerical Integration on the Sphere, pp. 2671–2710. Springer, Berlin (2015)CrossRefGoogle Scholar
  27. 27.
    Hickernell, F.J.: A generalized discrepancy and quadrature error bound. Math. Comput. 67(221), 299–322 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Mhaskar, H.N., Narcowich, F.J., Ward, J.D.: Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature. Math. Comput. 70(235), 1113–1130 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Müller, C.: Spherical Harmonics. Lecture Notes in Mathematics, vol. 17. Springer, Berlin (1966)zbMATHCrossRefGoogle Scholar
  30. 30.
    NIST Digital Library of Mathematical Functions., Release 1.0.10 of 2015-08-07
  31. 31.
    Olver, F.W.J.: Asymptotics and Special Functions. Computer Science and Applied Mathematics. Academic [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York (1974)Google Scholar
  32. 32.
    Owen, A.B.: The dimension distribution and quadrature test functions. Stat. Sin. 13(1), 1–17 (2003)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series. Special functions, vol. 2. Gordon & Breach Science Publishers, New York (1986). Translated from the Russian by N.M. QueenGoogle Scholar
  34. 34.
    Renka, R.J.: Multivariate interpolation of large sets of scattered data. ACM Trans. Math. Softw. 14, 139–148 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Sloan, I.H., Womersley, R.S.: Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math. 21(1–2), 107–125 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Sloan, I.H., Womersley, R.S.: A variational characterisation of spherical designs. J. Approx. Theory 159(2), 308–318 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Sloan, I.H., Womersley, R.S.: Filtered hyperinterpolation: a constructive polynomial approximation on the sphere. GEM Int. J. Geomath. 3(1), 95–117 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Stolarsky, K.B.: Sums of distances between points on a sphere. II. Proc. Am. Math. Soc. 41, 575–582 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Surjanovic, S., Bingham, D.: Virtual library of simulation experiments: test functions and datasets. Retrieved September 20, 2016, from
  40. 40.
    Szegő, G.: Orthogonal Polynomials. Colloquium Publications, vol. XXIII, 4th edn. American Mathematical Society, Providence (1975)Google Scholar
  41. 41.
    Ursell, F.: Integrals with nearly coincident branch points: Gegenbauer polynomials of large degree. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463(2079), 697–710 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Wang, Y.: Filtered polynomial approximation on the sphere. Bull. Aust. Math. Soc. 93(1), 162–163 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Wang, H., Wang, K.: Optimal recovery of Besov classes of generalized smoothness and Sobolev classes on the sphere. J. Complexity 32(1), 40–52 (2016)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Analysis and Number TheoryGraz University of TechnologyGrazAustria

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