Skip to main content
SpringerLink
Log in

Extremal functions of cubature formulas on a multidimensional sphere and spherical splines

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

Abstract

We establish the general form of extremal cubature formulas on multidimensional spheres. The domains of definition for the cubature formulas under consideration are Sobolev-type spaces on the sphere. The smoothness of the class function under study may be fractional. We prove that, for a given set of nodes, there exists a one-to-one correspondence between the set of extremal functions of cubature formulas on the sphere and the set of natural spherical splines with zero spherical mean.

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. M. Abramowitz (ed.) and I. A. Stegun (ed.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Appl. Math Series 55 (National Bureau of Standards, 1964).

  2. W. Freeden, M. Schreiner, and R. Franke, “A survey on spherical spline approximation”, Surveys Math. Indust. 7(1), 29–85 (1997)

    MathSciNet  MATH  Google Scholar 

  3. M. I. Ignatov and A. B. Pevnyi, Natural Splines of Several Variables (Nauka, Leningrad, 1991).

    Google Scholar 

  4. P. I. Lizorkin and S. M. Nikol’skiĭ, “Approximation by spherical functions”, Studies in the theory of differentiable functions of several variables and its applications, 11, Trudy Mat. Inst. Steklov 173, 181–189 (1986) [Proc. Steklov Inst.Math. 173, 195–203 (1987)].

    Google Scholar 

  5. S. G. Mikhlin, Multidimensional singular integrals and integral equations, (GIFML, Moscow, 1962) [Pergamon Press, Oxford-New York-Paris, 1965].

    Google Scholar 

  6. C. Müller, Spherical Harmonics (Springer-Verlag, Berlin, 1966).

    MATH  Google Scholar 

  7. S. L. Sobolev, Introduction to the Theory of Cubature Formulas (Nauka, Moscow, 1974) [in Russian].

    Google Scholar 

  8. S. L. Sobolev and V. L. Vaskevich, Cubature Formulas (Inst. Matem., Novosibirsk, 1996) [English translation: Kluwer Academic Publishers, Dordrecht and Boston, 1997].

    MATH  Google Scholar 

  9. H. Triebel, “Sampling numbers and embedding constants”, Tr. Mat. Inst. Steklov 248(1), 275–284 (2005) [Proc. Steklov Inst.Math. 248, 268–277 (2005)].

    MathSciNet  Google Scholar 

  10. V. L. Vaskevich, “A criterion for the guaranteed accuracy of the computation of multidimensional integrals”, Vychisl. Tekhnol. 9, Special Issue, 44–49 (2004) [in Russian].

    MATH  Google Scholar 

  11. V. L. Vaskevich, “On perturbations of the error for small perturbations of the weights of a cubature formula”, Vychisl. Tekhnol. 11, Special Issue, 19–26 (2006) [in Russian].

    MATH  Google Scholar 

  12. V. L. Vaskevich, “Embedding constants for periodic Sobolev spaces of fractional order”, Sibirsk.Mat. Zh. 49(5), 1019–1027 (2008) [SiberianMath. J. 49 (5), 806–813 (2008).

    MathSciNet  MATH  Google Scholar 

  13. V. L. Vaskevich, “Embedding constants and embedding functions for Sobolev-like spaces on the unit sphere”, Dokl. Akad. Nauk 433(4), 441–446 (2010) [Dokl.Math. 82 (1), 568–572 (2010)].

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

    V. L. Vaskevich

Authors
  1. V. L. Vaskevich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. L. Vaskevich.

Additional information

Original Russian Text © V. L. Vaskevich, 2011, published in Matematicheskie Trudy, 2011, Vol. 14, No. 2, pp. 14–27.

About this article

Cite this article

Vaskevich, V.L. Extremal functions of cubature formulas on a multidimensional sphere and spherical splines. Sib. Adv. Math. 22, 217–226 (2012). https://doi.org/10.3103/S1055134412030054

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1055134412030054

Keywords

Search

Navigation