If k is a positive integer an n-variate k-harmonic cardinal spline is a tempered distribution f on R n such that ∆ k f is a measure supported on the integer lattice Z n in R n . Symbolically
$${{\Delta }^{k}}f\left( x \right)=\sum\limits_{\text{j}\in {{\text{Z}}^{n}}}{{{a}_{j}}}\delta \left( x-j \right)$$
where ∆ is the n variate Laplacian, ∆k = ∆∆k−1 for k > 1, and δ(x) is the unit Dirac measure supported at the origin. A polyharmonic spline is one which is k-harmonic for some k.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. H. Ahlberg, E. N. Nilson, and J. L. Walsh, The Theory of Splines and Their Applications, Academic Press, New York, 1967.Google Scholar
  2. [2]
    M. D. Buhmann and M. J. D. Powell, Radial basis function interpolation on an infinite regular grid, preprint 1988, 24 p.Google Scholar
  3. [3]
    C. K. Chui, Multivariate Splines, CBMS Vol. 54, SIAM, Philadelphia, 1988.Google Scholar
  4. [4]
    C. deBoor, How small can one make the derivatives of an interpolating function?, J. Approx. Theory 13 (1975), 105–116.CrossRefGoogle Scholar
  5. [5]
    C. de Boor, K. Hollig, and S. Riemenschnieder, Bivariate cardinal interpolation on a three-direction mesh, Ill. J. Math., 29 (1985), 533–566.Google Scholar
  6. [6]
    J. Duchon, Splines minimizing rotation-invariant seminorms in Sobolev spaces, in Constructive Theory of Functions of Several Variables, W. Schempp and K. Zeller eds., Springer-Verlag, 1977, 85–100.CrossRefGoogle Scholar
  7. [7]
    N. Dyn, Survey Lecture, to appear in Approximation Theory VI, C. Chui, L. Schumaker, J. Ward, eds., Academic Press, New York, 1989.Google Scholar
  8. [8]
    M. Golomb and H. F. Weinberger, Optimal approximation and error bounds, in On Numerical Approximation, R. E. Langer, ed., Madison, 1959, 117–190.Google Scholar
  9. [9]
    W. R. Madych, Polyharmonic splines, unpublished preprint 1978, 16 p.Google Scholar
  10. [10]
    W. R. Madych and S. A. Nelson, Polyharmonic cardinal splines, Abstracts Amer. Math. Soc. 7 (1986), 378.Google Scholar
  11. [11]
    W. R. Madych and S. A. Nelson, Polyharmonic cardinal splines, to appear in J. Approx. Theory.Google Scholar
  12. [12]
    W. R. Madych and S. A. Nelson, Polyharmonic cardinal splines: a minimization property, technical report, BRC/MATH-TR-89–1.Google Scholar
  13. [13]
    W. R. Madych and S. A. Nelson, Multivariate interpolation and conditionally positive definite functions II, to appear in Math. Comp.Google Scholar
  14. [14]
    W. R. Madych, Splines and entire functions, technical report, BRC/MATH-TR-89–2.Google Scholar
  15. [15]
    J. Meinguet, Multivariate interpolation at arbitrary points made simple, Z. Angew Math. Phys., 30 (1979), 292–304.CrossRefGoogle Scholar
  16. [16]
    C. A. Micchelli, Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx., 2 (1986), 11–22.CrossRefGoogle Scholar
  17. [17]
    E. H. Potter, Multivariate Polyharmonic Spline Interpolation, Ph.D. thesis, Iowa State University, 1981.Google Scholar
  18. [18]
    W. Schempp, Complex Contour Integral Representation of Cardinal Spline Functions, Contemporary Mathematics, Vol. 7, Amer. Math. Soc., Providence, 1982.CrossRefGoogle Scholar
  19. [19]
    I. J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, Quart. Appl. Math., 4 (1946), 45–99.Google Scholar
  20. [20]
    I. J. Schoenberg, Cardinal Spline Interpolation, CBMS Vol. 12, SIAM, Philadelphia, 1973.Google Scholar
  21. [21]
    G. Wahba, Spline bases, regularization, and generalized cross validation for solving approximation problems with large quantities of noisy data, Approximation Theory III, E. W. Cheney, ed., Academic Press, New York, 1980.Google Scholar

Copyright information

© Birkhäuser Verlag Basel 1989

Authors and Affiliations

  • W. R. Madych
    • 1
  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA

Personalised recommendations