Numerical Algorithms

, Volume 39, Issue 1–3, pp 307–315 | Cite as

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



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).


interpolation scattered data radial basis functions band-limited functions error estimates 


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  1. [1]
    R.A. Brownlee and W. Light, Approximation orders for interpolation by surface splines to rough functions, IMA J. Numer. Anal., to appear. Google Scholar
  2. [2]
    M.D. Buhmann, Multivariate cardinal interpolation with radial basis functions, Constr. Approx. 6 (1990) 225–255. Google Scholar
  3. [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. [4]
    M.D. Buhmann, Radial Basis Functions: Theory and Implementations (Cambridge Univ. Press, Cambridge, 2003). Google Scholar
  5. [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. [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. [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. [8]
    P. Erdős, On some convergence properties of the interpolation polynomials, Ann. Math. 44 (1943) 330–337. Google Scholar
  9. [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. [10]
    D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, Berlin, 1977). Google Scholar
  11. [11]
    M. Johnson, A note on the limited stability of surface splines, Preprint (2002). Google Scholar
  12. [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. [13]
    W.R. Madych and S.A. Nelson, Multivariate interpolation and conditionally positive definite functions II, Math. Comp. 54 (1990) 211–230. MATHGoogle Scholar
  14. [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. [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. CrossRefGoogle Scholar
  16. [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. Google Scholar
  17. [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. Google Scholar
  18. [18]
    R. Schaback, Approximation by radial basis functions with finitely many centers, Constr. Approx. 12 (1996) 331–340. Google Scholar
  19. [19]
    R. Schaback, Improved error bounds for scattered data interpolation by radial basis functions, Math. Comp. 68 (1999) 201–216. Google Scholar
  20. [20]
    E.M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton Univ. Press, Princeton, NJ, 1971). Google Scholar
  21. [21]
    H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, AICM 4 (1995) 389–396. Google Scholar
  22. [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. [23]
    H. Wendland, Meshless Galerkin methods using radial basis functions, Math. Comp. 68 (1999) 1521–1531. Google Scholar
  24. [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. [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

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© Springer 2005

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA

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