The first part of this paper is concerned with the numerical evaluation of multivariate B-splines (cf. [7]) by means of certain recurrence relations [11],[4] involving only convex combinations of positive quantities. In the second part facilities of approximating by linear combinations of such B-splines are discussed. In particular, this leads to the construction of linear approximation schemes providing good approximations for a given function as well as for its derivatives of any order (if they exist) lower than the degree of the splines.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Allgower, E. und Georg, K.: Triangulations by reflections with applications to approximation. Numerische Methoden der Approximationstheorie Band 3, ISNM Birkhäuser, Basel-Stuttgart, (1978),Google Scholar
  2. 2.
    Bramble, J.H. und Zlarnal, M.: Triangular elements in the finite element method. Math. Comp. 24(1970), 809–820.CrossRefGoogle Scholar
  3. 3.
    Cox, M.G.: The numerical evaluation of B-splines. J. Inst. Maths. Applics. 10 (1972), 134–149.CrossRefGoogle Scholar
  4. 4.
    Dahmen, W.: On multivariate B-splines. Erscheint in SIAM J. Numer. Anal.Google Scholar
  5. 5.
    Dahmen, W.: Polynomials as linear combinations of multivariate B-splines. Eingereicht bei Math. Zeitschrift.Google Scholar
  6. 6.
    Dahmen, W.: Approximation by linear combinations of multivariate B-splines. In Vorbereitung.Google Scholar
  7. 7.
    de Boor, C: Splines as linear combinations of B-splines. A survey in Approximation Theory II, Edited by G.G. Lorentz, C.K. Chui, L.L. Schumaker, Academic Press, 1976, 1–47.Google Scholar
  8. 8.
    de Boor, C. und Fix, G.: Spline approximation by quasiinterpolants. J. Approximation Theory 7 (1973), 19–45.CrossRefGoogle Scholar
  9. 9.
    Kuhn, H.W.: Some combinatorial lemmas in topology. IBM J. Research and Develop. 45 (1960), 518–524.CrossRefGoogle Scholar
  10. 10.
    Lyche, T. und Schumaker, L.L.: Local spline approximation methods. J. Approximation Theory 15 (1975), 294–325.CrossRefGoogle Scholar
  11. 11.
    Micchelli, C.A.: A constructive approach to Kergin interpolation in Rk: Multivariate B-splines and Lagrange interpolation. MRC Technical Summary Report, 1978.Google Scholar
  12. 12.
    Morrey, C.B.: Multiple integrals in the calculus of variations. Berlin — Heidelberg — New York, Springer, 1966.Google Scholar

Copyright information

© Springer Basel AG 1980

Authors and Affiliations

  • Wolfgang Dahmen
    • 1
  1. 1.Institut für Angewandte MathematikUniversität BonnBonnDeutschland

Personalised recommendations