On the Structure of Kergin Interpolation for Points in General Position

  • Len Bos
  • Shayne Waldron
Conference paper
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 137)


For n+1 points in ℝd, in general position, the Kergin polynomial interpolant of C n functions may be extended to an interpolant of C d−1functions. This results in an explicit set of reduced Kergin functionals naturally stratified by their dependence on certain directional derivatives of order k,0 ≤ kd−1. We show that the polynomials dual to the functionals depending on derivatives of order k are multi-ridge functions of d−k variables and moreover, that the polynomials dual to the purely interpolating functionals(k =0)are always harmonic.


General Position Point Evaluation Directional Derivative Lagrange Interpolation Harmonic Polynomial 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Andersson, M. Passare: Complex Kergin interpolation, J. Approx. Theory, 64 (1991), 214–225.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    L. Bos, J.-P. Calvi: Kergin interpolants at the roots of unity approximate C 2 functions, J. Anal. Math. 72 (1997), 203–221.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    T. N. T. Goodman: Interpolation in minimum semi-norm,and multivariate B-splines,J. Approx. Theory 37 (1983), 212–223.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    P. Kergin: A natural interpolation of C k functions, J. Approx. Theory 29 (1980), 278–293.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    V. Ya. Lin, A. Pinkus: Fundamentality of ridge functions, J. Apptox. Theory 75(3) (1993), 295–311.zbMATHCrossRefGoogle Scholar
  6. [6]
    C. A. Micchelli: A constructive approach to Kergin interpolation ink:multivariate B-splines and Lagrange interpolation, Rocky Mountain J. Math. 10 (1980), 485–497.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    C. A. Micchelli, P. Milman: A formula for Kergin interpolation ink, J. Approx. Theory 29 (1980), 294–296.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    S. Waldron: Mean value interpolation for points in general position, preprintGoogle Scholar

Copyright information

© Springer Basel AG 2001

Authors and Affiliations

  • Len Bos
    • 1
  • Shayne Waldron
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryCalgary, AlbertaCanada
  2. 2.Department of MathematicsUniversity of AucklandAucklandNew Zealand

Personalised recommendations