Quasi-Interpolation on Irregular Points

  • E. W. Cheney
  • Junjiang Lei
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 119)


A quasi-interpolant is an operator L having the form
$$Lf = \sum\limits_{i = 1}^\infty {f\left( {{y_i}} \right){g_i}} .$$
The points y i are called “nodes”; they are prescribed in ℝ n . The entities g i are prescribed functions from ℝ n to ℝ. The case of irregularly situated nodes is of particular interest. We investigate the question of how to select the “base functions” g i in order to obtain favorable estimates of ∥Lf - f∥.


Radial Basis Function General Position Lagrange Interpolation Lagrange Polynomial Vandermonde Determinant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Beatson R. K., Light W. A. Quasi-interpolation in the absence of polynomial reproduction. Pre-print, May 1992.Google Scholar
  2. [2]
    Beatson R. K., Powell M. J. D. Univariate multiquadric approximation: quasi-interpolation to scattered data. Report DAMPT-NA7, Cambridge University, 1990.Google Scholar
  3. [3]
    de Boor C. Quasi-interpolants and approximation power of multivariate splines. In Computation of Curves and Surfaces, pages 313–345. Kluwer, 1990. W. Dahmen, M. Gasca and C. A. Micchelli, eds.CrossRefGoogle Scholar
  4. [4]
    de Boor C. Approximation order without quasi-interpolants. In Approximation Theory VII, pages 1–18. Academic Press, New York, 1993. E. W. Cheney, C. K. Chui, and L. L. Schumaker, eds.Google Scholar
  5. [5]
    de Boor C. The quasi-interpolant as a tool in elementary polynomial spline theory. In Approximation Theory, pages 269–276. Academic Press, New York, 1973. G. G. Lorentz, ed.Google Scholar
  6. [6]
    de Boor C., Fix G. Spline approximation by quasi-interpolants. J. Approx. Theory, 8: 19–45, 1973.zbMATHCrossRefGoogle Scholar
  7. [7]
    de Boor C., Jia R. Q. Controlled approximation and a characterization of the local approximation order. Proc. Amer. Math. Soc., 95: 547–553, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Bos L. A characteristic of points in R2 having Lebesgue function of polynomial growth. J. Approx. Theory, 56: 316–329, 1989.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Bos L. Some remarks on the Fejér problem for Lagrange interpolation in several variables. J. Approx. Theory, 60: 133–140, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Buhmann M. D. Convergence of univariate quasi-interpolation using multi-quadrics. IMA J. Numer. Anal., 8: 365–384, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Buhmann M. D. On quasi-interpolation with radial basis functions. J. Approx. Theory, 72: 103–130, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Buhmann M. D., Dyn N., Levin D. On quasi-interpolation with radial basis functions on non-regular grids. Submitted to Constr. Approx..Google Scholar
  13. [13]
    Chui C. K., Diamond H. A natural formulation of quasi-interpolation by multivariate splines. Proc. Amer. Math. Soc., 99: 643–646, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Chui C. K., Shen X. C., Zhong L. On Lagrange polynomial quasi-interpolation. In Topics in Polynomials of one and Several Variables and their Applications, pages 125–142. World Scientific Publishers, Singapore, 1993. T. H. Rassias et al., eds.Google Scholar
  15. [15]
    Frederickson P. O. Quasi-interpolation, extrapolation, and approximation in the plane. In Proceedings of the Manitoba Conference on Numerical Mathematics, pages 159–167. Dept. Comput. Sci., University of Manitoba, 1971.Google Scholar
  16. [16]
    Jia R. Q., Lei J. Approximation by multi-integer translates of functions having global support. J. Approx. Theory, 72: 2–23, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    Light W. A., Cheney E. W. Quasi-interpolation with translates of a function having non-compact support. Constr. Approx., 8: 35–48, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Rabut C. How to build quasi-interpolants with application to polyharmonic B-splines. In Curves and Surfaces. Academic Press, New York, 1991. P. J. Laurent, A. Le Méhauté and L. L. Schumaker, eds.Google Scholar
  19. [19]
    Rabut C. B-Splines polyharmoniques cardinales: interpolation, quasi-interpolation, filtrage. Thesis, Toulouse, 1990.Google Scholar
  20. [20]
    Schoenberg I. J. Contributions to the problem of approximation of equidistant data by analytic functions. Quart. Appl. Math., 4:45–99, 112–141, 1946.MathSciNetGoogle Scholar
  21. [21]
    Smith P. W., Ward J. D. Quasi-interpolants from spline interpolation operators. Constr. Approx., 6: 97–110, 1990.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser 1994

Authors and Affiliations

  • E. W. Cheney
    • 1
  • Junjiang Lei
    • 1
  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA

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