Skip to main content
SpringerLink
Log in

Recent developments in error estimates for scattered-data interpolation via radial basis functions

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Error estimates for scattered data interpolation by “shifts” of a positive definite function for target functions in the associated reproducing kernel Hilbert space (RKHS) have been known for a long time. However, apart from special cases where data is gridded, these interpolation estimates do not apply when the target functions generating the data are outside of the associated RKHS, and in fact until very recently no estimates were known in such situations. In this paper, we review these estimates in cases where the underlying space is Rn and the positive definite functions are radial basis functions (RBFs).

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R.A. Brownlee and W. Light, Approximation orders for interpolation by surface splines to rough functions, IMA J. Numer. Anal., to appear.

  2. M.D. Buhmann, Multivariate cardinal interpolation with radial basis functions, Constr. Approx. 6 (1990) 225–255.

    Google Scholar 

  3. M.D. Buhmann, New developments in the theory of radial basis function interpolation, in: Multivariate Approximation: From CAGD to Wavelets, Santiago (1992), eds. K. Jetter and F.I. Utreras (World Scientific, Singapore, 1993) pp. 35–75.

    Google Scholar 

  4. M.D. Buhmann, Radial Basis Functions: Theory and Implementations (Cambridge Univ. Press, Cambridge, 2003).

    Google Scholar 

  5. C. de Boor, R. DeVore and A. Ron, Approximation from shift-invariant subspaces of L2(Rd), Trans. Amer. Math. Soc. 341 (1994) 787–806.

    Google Scholar 

  6. J. Duchon, Splines minimizing rotation invariant semi-norms in Sobolev spaces, in: Constructive Theory of Functions of Several Variables, Proc. of Conf., Math. Res. Inst., Oberwolfach (1976), Lecture Notes in Mathematics, Vol. 571 (Springer, Berlin, 1977) pp. 85–100.

    Google Scholar 

  7. J. Duchon, Sur l’erreur d’interpolation des fonctions de plusieurs variables par les Dm-splines, RAIRO Anal. Numér. 12(4) (1978) vi 325–334.

    Google Scholar 

  8. P. Erdős, On some convergence properties of the interpolation polynomials, Ann. Math. 44 (1943) 330–337.

    Google Scholar 

  9. M. Frazier, B. Jawerth and G. Weiss, Littlewood–Paley Theory and the Study of Function Spaces, CBMS Regional Conference Series in Mathematics, Vol. 79 (Amer. Math. Soc., Providence, RI, 1991).

    Google Scholar 

  10. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, Berlin, 1977).

    Google Scholar 

  11. M. Johnson, A note on the limited stability of surface splines, Preprint (2002).

  12. W.R. Madych and S.A. Nelson, Multivariate interpolation and conditionally positive definite functions, Approx. Theory Appl. 4 (1988) 77–79.

    Google Scholar 

  13. W.R. Madych and S.A. Nelson, Multivariate interpolation and conditionally positive definite functions II, Math. Comp. 54 (1990) 211–230.

    MATH  Google Scholar 

  14. H.N. Mhaskar, F.J. Narcowich, N. Sivakumar and J.D. Ward, Approximation with interpolatory constraints, Proc. Amer. Math. Soc. 130 (2002) 1355–1364.

    Google Scholar 

  15. F.J. Narcowich and J.D. Ward, Scattered-data interpolation on spheres: Error estimates and locally supported basis functions, SIAM J. Math. Anal. 33 (2002) 1393–1410.

    Article  Google Scholar 

  16. F.J. Narcowich and J.D. Ward, Scattered-data interpolation on Rn: Error estimates for radial basis and band-limited functions, SIAM J. Math. Anal., to appear.

  17. F.J. Narcowich, J.D. Ward and H. Wendland, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Math. Comp., to appear.

  18. R. Schaback, Approximation by radial basis functions with finitely many centers, Constr. Approx. 12 (1996) 331–340.

    Google Scholar 

  19. R. Schaback, Improved error bounds for scattered data interpolation by radial basis functions, Math. Comp. 68 (1999) 201–216.

    Google Scholar 

  20. E.M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton Univ. Press, Princeton, NJ, 1971).

    Google Scholar 

  21. H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, AICM 4 (1995) 389–396.

    Google Scholar 

  22. H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory 93 (1998) 258–272.

    Google Scholar 

  23. H. Wendland, Meshless Galerkin methods using radial basis functions, Math. Comp. 68 (1999) 1521–1531.

    Google Scholar 

  24. Z. Wu and R. Schaback, Local error estimates for radial basis function interpolation of scattered data, IMA J. Numer. Anal. 13 (1993) 13–27.

    Google Scholar 

  25. J. Yoon, Spectral approximation orders of radial basis function interpolation on the Sobolev space, SIAM J. Math. Anal. 33 (2001) 946–958.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Francis J. Narcowich

Authors
  1. Francis J. Narcowich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francis J. Narcowich.

Additional information

AMS subject classification

41A25, 41A05, 41A63, 42B35

Research supported by grant DMS-0204449 from the National Science Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Narcowich, F.J. Recent developments in error estimates for scattered-data interpolation via radial basis functions. Numer Algor 39, 307–315 (2005). https://doi.org/10.1007/s11075-004-3644-7

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-004-3644-7

Keywords

Search

Navigation