Advances in Computational Mathematics

, Volume 9, Issue 3–4, pp 251–279 | Cite as

On the approximation power of bivariate splines

  • Ming-Jun Lai
  • Larry L. Schumaker


We show how to construct stable quasi-interpolation schemes in the bivariate spline spaces Sdr(Δ) with d⩾ 3r + 2 which achieve optimal approximation order. In addition to treating the usual max norm, we also give results in the Lp norms, and show that the methods also approximate derivatives to optimal order. We pay special attention to the approximation constants, and show that they depend only on the smallest angle in the underlying triangulation and the nature of the boundary of the domain.

bivariate splines approximation order by splines stable approximation schemes super-splines 41A15 41A63 41A25 65D10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. Adams, Sobolev Spaces (Academic Press, New York, 1975).MATHGoogle Scholar
  2. [2]
    P. Alfeld, M. Neamtu and L.L. Schumaker, Dimension and local bases of homogeneous spline spaces, SIAM J. Math. Anal. 27 (1996) 1482–1501.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    P. Alfeld and L.L. Schumaker, The dimension of bivariate spline spaces of smoothness r for degree d ⩾ 4r + 1, Constr. Approx. 3 (1987) 189–197.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    C. de Boor, B-form basics, in: Geometric Modeling: Algorithms and New Trends, ed. G.E. Farin (SIAM, Philadelphia, PA, 1987) pp. 131–148.Google Scholar
  5. [5]
    C. de Boor and K. Höllig, Approximation power of smooth bivariate pp functions, Math. Z. 197 (1988) 343–363.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods (Springer, New York, 1994).MATHGoogle Scholar
  7. [7]
    C.K. Chui, D. Hong and R.-Q. Jia, Stability of optimal order approximation by bivariate splines over arbitrary triangulations, Trans. Amer. Math. Soc. 347 (1995) 3301–3318.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    C.K. Chui and M.J. Lai, On bivariate super vertex splines, Constr. Approx. 6 (1990) 399–419.MATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    G. Farin, Triangular Bernstein–Bézier patches, Comput. Aided Geom. Design 3 (1986) 83–127.MathSciNetCrossRefGoogle Scholar
  10. [10]
    D. Hong, Spaces of bivariate spline functions over triangulation, Approx. Theory Appl. 7 (1991) 56–75.MATHGoogle Scholar
  11. [11]
    A. Ibrahim and L.L. Schumaker, Super spline spaces of smoothness r and degree d ⩾ 3r + 2, Constr. Approx. 7 (1991) 401–423.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    R.Q. Jia, Approximation order from certain spaces of smooth bivariate splines on a three-direction mesh, Trans. Amer. Math. Soc. 295 (1986) 199–212.MATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    L.L. Schumaker, Spline Functions: Basic Theory (Wiley, New York, 1981).Google Scholar
  14. [14]
    L.L. Schumaker, On super splines and finite elements, SIAM J. Numer. Anal. 26 (1989) 997–1005.MATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    E.M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, NJ, 1970).MATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Ming-Jun Lai
    • 1
  • Larry L. Schumaker
    • 2
  1. 1.Department of MathematicsThe University of GeorgiaAthensUSA
  2. 2.Department of MathematicsVanderbilt UniversityNashvilleUSA

Personalised recommendations