Skip to main content
Log in

Interpolation of scattered data: Distance matrices and conditionally positive definite functions

  • Published:
Constructive Approximation Aims and scope


Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R. Franke.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. M. Abramowitz, I. A. Stegun (1966): Handbook of Mathematical Functions (National Bureau of Standards Applied Mathematics Series 55). Washington, D.C.: National Bureau of Standards.

    Google Scholar 

  2. F. P. Agterberg (1974): Geomathematics. Amsterdam: Elsevier.

    Google Scholar 

  3. N. Aronszajn (1950):Theory of reproducing kernels. Trans. Amer. Math. Soc.,68:337–404.

    Google Scholar 

  4. R. Askey (1973): Radial Characteristic Functions (MRC Technical Sum: Report no. 1262). University of Wisconsin.

  5. R. Barnhill, S. Stead (1983):Multistage trivariate surfaces. Preprint.

  6. Blumenthal (1970): Theory and Application of Distance Geometry. New York: Chelsea Publishing.

    Google Scholar 

  7. W. F. Donoghue (1974): Monotone Matrix Functions and Analytic Continuation. Berlin Heidelberg New York: Springer-Verlag.

    Google Scholar 

  8. J. Duchon (1976):Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In: Constructive Theory of Functions of Several Variables (W. Schempp, K. Zeller, eds.). Berlin Heidelberg New York: Springer-Verlag.

    Google Scholar 

  9. N. Dyn, D. Levin, S. Rippa (1983):Numerical procedures for global surface smoothing of noisy scattered data. Preprint.

  10. R. Franke (1982):Scattered data interpolation: tests of some methods. Math. Comp.,38:381–200.

    Google Scholar 

  11. R. Franke (1983):Lecture notes on global basis function methods for scattered data. International Symposium on Surface Approximation, University of Milano, Govgano, Italy.

    Google Scholar 

  12. G. Gasper (1975):Positivity and special functions. In: The Theory and Application of Special Functions (R. Askey, ed.). New York: Academic Press.

    Google Scholar 

  13. I. M. Gelfand, N. Ya. Vilenkin (1964): Generalized Functions, Vol. 4. New York: Academic Press.

    Google Scholar 

  14. R. L. Hardy (1971):Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res., C.

  15. R. L. Hardy (1982):Surface fitting with biharmonic and harmonic models. In: Proceedings of the NASA Workshop on Surface Fitting. College Station, Texas: Center for Approximation Theory, Texas A & M University, pp. 136–146.

    Google Scholar 

  16. S. Helgason (1980): The Radon Transform. Basel: Birkhäuser.

    Google Scholar 

  17. J. V. Linnik (1953):Linear forms and statical criteria, II. Ukrain. Mat. Zh.,51:247–290 [also (1962): Selected Translations in Mathematical Statistics and Probability, Vol. 5. Providence: American Mathematical Society, pp. 41–90].

    Google Scholar 

  18. E. Luckas (1970): Characteristic Functions. New York: Hafner.

    Google Scholar 

  19. G. Matheron (1973):The intrinsic random functions and their applications. Adv. in Appl. Probab.,5:439–468.

    Google Scholar 

  20. G. Matheron (1981):Splines and kriging: their formal equivalence. In: Syracuse University Geology Contribution 8 (D. F. Marriam, ed.). Syracuse, New York: Department of Geology, Syracuse University.

    Google Scholar 

  21. J. Meinguet (1979):An intrinsic approach to multivariate spline interpolation at arbitrary points. In: Polynomial and Splines Approximation (B. N. Sahney, ed.). Dordrecht: D. Reidel, pp. 163–190.

    Google Scholar 

  22. J. von Neumann, I. J. Schoenberg (1941):Fourier integrals and metric geometry. Trans. Amer. Math. Soc.,50:226–251.

    Google Scholar 

  23. K. Salkauskas (1982):Some relationships between surface splines and kriging. In: Multivariate Approximation Theory II (W. Schempp, K. Zeller, eds.). Basel: Birkhäuser, pp. 313–325.

    Google Scholar 

  24. I. P. Schagen (1979):Interpolation in two dimensions — a new technique. J. Inst. Math. Appl.,23:53–59.

    Google Scholar 

  25. I. J. Schoenberg (1935):Remarks to Maurice Frechet's article “Sur la definition axiomatique d'une classe d'espace distancies vectoriellemment applicable sur l'espace de Hilbert.” Ann. of Math.,36:724–732.

    Google Scholar 

  26. I. J. Schoenberg (1938):Metric spaces and positive definite functions. Trans. Amer. Math. Soc.,44:522–536.

    Google Scholar 

  27. I. J. Schoenberg (1938):Metric spaces and completely monotone functions. Ann. of Math.,39:811–841.

    Google Scholar 

  28. J. Stewart (1976):Positive definite functions and generalizations, an historical survey. Rocky Mountain J. Math.,6:409–434.

    Google Scholar 

  29. Wahba (1982): Private communication.

  30. G. N. Watson (1966): Theory of Bessel Functions, 2nd ed. Cambridge: Cambridge University Press.

    Google Scholar 

  31. D. V. Widder (1946): The Laplace Transform. Princeton: Princeton University Press.

    Google Scholar 

  32. R. E. Williamson (1956):Multiply monotone functions and their Laplace transform. Duke Math. J.,23:189–207.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Additional information

Communicated by Carl de Boor.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Micchelli, C.A. Interpolation of scattered data: Distance matrices and conditionally positive definite functions. Constr. Approx 2, 11–22 (1986).

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI:

AMS classification

Key words and phrases