On Some Multivariate Quadratic Spline Quasi—Interpolants on Bounded Domains

  • Paul Sablonnière
Conference paper
Part of the International Series of Numerical Mathematics book series (ISNM, volume 145)


We study some C1quadratic (or d—quadratic) spline quasi—interpolants on bounded domains \(\Omega \subset I{R^d},d = 1,2,3.\ \)


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. Barrera, M. J. Ibanéz, P. Sablonnière: Near-best quasi-interpolants on uniform and nonuniform partitions in one and two dimensions. Prépublication IRMAR 02–52, 2002. Submitted for publication in Proceedings of the International Conference on Curves and Surfaces (Saint-Malo, July 2002).Google Scholar
  2. 2.
    G. Baszenski: n-th order polynomial spline blending, In: Multivariate Approximation Theory III, W. Schempp, K. Zeller (eds.), ISNM, Volume 75, 35–46, Birkhäuser, Basel 1985.CrossRefGoogle Scholar
  3. 3.
    L. Beutel, H. Gonska: Simultaneous approximation by tensor product operators, Technical report SM-DU-504, Universität Duisburg, 2001.Google Scholar
  4. 4.
    B. D. Bojanov, H. A. Hakopian, A. A. Sahakian: Spline Functions and Multivariate Interpolation, Kluwer, Dordrecht 1993.CrossRefGoogle Scholar
  5. 5.
    C. de Boor: A Practical Guide to Splines, Springer, New York 2001. (revised edition).zbMATHGoogle Scholar
  6. 6.
    C. de Boor: Splines as linear combinations of B-splines,In: Approximation Theory II, G.G. Lorentz et al. (eds.), 1–47, Academic Press, New York 1976.Google Scholar
  7. 7.
    C. de Boor: Quasiinterpolants and approximation power of multivariate splines, In: Computation of Curves and Surfaces, W. Dahmen, M. Gasca, C. A. Micchelli (eds.), 313–345, Kluwer, Dordrecht 1990.Google Scholar
  8. 8.
    C. de Boor, K. Höllig, S. Riemenschneider: Box-splines, Springer, New York 1993.CrossRefzbMATHGoogle Scholar
  9. 9.
    C. de Boor, G. Fix: Spline approximation by quasi-interpolants, J. Approx. Theory 8 (1973), 19–45.CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    G. Chen, C. K. Chui, M. J. Lai: Construction of real-time spline quasi-interpolation schemes, CAT Report 107, Texas A&M University, 1986.Google Scholar
  11. 11.
    C. K. Chui, M. J. Lai: A multivariate analog of Marsden’s identity and a quasi-interpolation scheme, Constr. Approx. 3 (1987), 111–122.zbMATHMathSciNetGoogle Scholar
  12. 12.
    C. K. Chui, L. L. Schumaker, R. H. Wang: On spaces of piecewise polynomials with boundary conditions III. Type II triangulations, In: Canadian Mathematical Society Conference Proceedings, Volume 3 (1983), 67–80, American Mathematical Society. MathSciNetGoogle Scholar
  13. 13.
    C. K. Chui: Multivariate Splines, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 54, SIAM, Philadelphia 1988.Google Scholar
  14. 14.
    F. J. Delvos, W. Schempp: Boolean Methods in Interpolation and Approximation, Longman Scientific & Technical 1989.Google Scholar
  15. 15.
    R. A. DeVore, G. G. Lorentz: Constructive Approximation, Springer, Berlin 1993.CrossRefzbMATHGoogle Scholar
  16. 16.
    B. Fornberg: A Practical Guide to Pseudospectral Methods, Cambridge University Press 1996.Google Scholar
  17. 17.
    H. Gonska: Degree of simultaneous approximation of bivariate functions by Gordon operators, J. Approx. Theory 62 (1990), 170–191. zbMATHMathSciNetGoogle Scholar
  18. 18.
    H. Gonska: Simultaneous approximation by generalized n-th order blending operators, In: ISNM Volume 90 173–180, Birkhäuser, Basel 1989. MathSciNetGoogle Scholar
  19. 19.
    W. J. Gordon: Distributive lattices and approximation of multivariate functions, In: Approximation with Special Emphasis on Spline Functions, I. J. Schoenberg (ed.), 223–277, Academic Press, New York 1969. Google Scholar
  20. 20.
    M. J. Ibañez-Pérez: Cuasi-interpolantes spline discretos con norma casi minima: teoria y aplicaciones. Tesis doctoral, Universidad de Granada, 2003 (In preparation).Google Scholar
  21. 21.
    T. Lyche, L. L. Schumaker: Local spline approximation, J. Approx. Theory 15 (1975), 294–325.CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    J. M. Marsden, I. J. Schoenberg: An identity for spline functions with applications to variation diminishing spline approximation. J. Approx. Theory 3 (1970), 7–49.CrossRefzbMATHGoogle Scholar
  23. 23.
    J. M. Marsden: Operator norm bounds and error bounds for quadratic spline interpolation, In: Approximation Theory, Banach Center Publications, vol. 4 (1979), 159–175.MathSciNetGoogle Scholar
  24. 24.
    G. Nürnberger: Approximation by Spline Functions, Springer, Berlin 1989.Google Scholar
  25. 25.
    M. J. D. Powell: Piecewise quadratic surface fitting in contour plotting, In: Software for Numerical Mathematics, D. J. Evans (ed.), 253–271. Academic Press, London 1974.Google Scholar
  26. 26.
    Rong-Qing Jia: Approximation by multivariate splines: an application of boolean methods, In: Numerical Methods of Approximation Theory, Volume 9, D. Braess, L. L. Schumaker (eds.), ISNM Volume 105, 117–134, Birkhäuser, Basel 1992.Google Scholar
  27. 27.
    P. Sablonnière: Bases de Bernstein et approximants spline. Thèse de doctorat, Université de Lille, 1982.Google Scholar
  28. 28.
    P. Sablonnière: Bernstein-Bézier methods for the construction of bivariate spline approximants. Comput. Aided Geom. Design 2 (1985), 29–36.CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    P. Sablonnière: Quasi-interpolants associated with H-splines on a three-direction mesh. J. Comput. & Appl. Math. 66 (1996), 433–442.CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    P. Sablonnière: Quasi-interpolantes sobre particiones uniformes, Meeting in Approximation Theory, Ubeda, Spain, July 2000. Prépublication IRMAR 00–38, 2000.Google Scholar
  31. 31.
    P. Sablonnière: H-splines and quasi-interpolants on a three directional mesh. Prépublication IRMAR 02–11, 2002. To appear in Proceedings IDoMAT 2001.Google Scholar
  32. 32.
    P. Sablonnière: BB-coefficients of basic bivariate quadratic splines on uniform and nonuniform criss-cross triangulations of a rectangular domain. Prépublications IRMAR, 2002 (to appear).Google Scholar
  33. 33.
    P. Sablonnière, F. Jeeawock-Zedek: Hermite and Lagrange interpolation by quadratic splines on nonuniform criss-cross triangulations. In: Curves and Surfaces P. J. Laurent, A. Le Méhauté, L. L. Schumaker (eds.)445–452, A.K. Peters, Wellesley 1991Google Scholar
  34. 34.
    I. J. Schoenberg: Cardinal Spline Interpolation, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 12, SIAM, Philadelphia 1973.Google Scholar
  35. 35.
    I. J. Schoenberg: Selected Papers, Volumes 1 and 2, edited by C. de Boor, Birkhäuser, Boston 1988.Google Scholar
  36. 36.
    L. L. Schumaker: Spline Functions: Basic Theory, John Wiley & Sons, New York 1981.zbMATHGoogle Scholar
  37. 37.
    L. N. Trefethen: Spectral Methods in Matlab, SIAM, Philadelphia 2000.CrossRefGoogle Scholar
  38. 38.
    J. Ward: Polynomial reproducing formulas and the commutator of a locally supported spline, In: Topics in multivariate approximation, C. K. Chui, L. L. Schumaker, F. Utreras (eds.), 255–263, Academic Press, Boston 1987.Google Scholar

Copyright information

© Springer Basel AG 2003

Authors and Affiliations

  • Paul Sablonnière
    • 1
  1. 1.INSA de RennesRennes cedexFrance

Personalised recommendations