Numerische Mathematik

, Volume 101, Issue 4, pp 729–748 | Cite as

Approximate Interpolation with Applications to Selecting Smoothing Parameters

  • Holger WendlandEmail author
  • Christian Rieger


In this paper, we study the global behavior of a function that is known to be small at a given discrete data set. Such a function might be interpreted as the error function between an unknown function and a given approximant. We will show that a small error on the discrete data set leads under mild assumptions automatically to a small error on a larger region. We will apply these results to spline smoothing and show that a specific, a priori choice of the smoothing parameter is possible and leads to the same approximation order as the classical interpolant. This has also a surprising application in stabilizing the interpolation process by splines and positive definite kernels.

Mathematics Subject Classification (2000)

65D10 65D07 41A25 


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  1. 1.
    Anselone, P.M., Laurent, P.J.: A general method for the construction of interpolating or smoothing spline-functions. Numer. Math. 12, 66–82 (1968)CrossRefGoogle Scholar
  2. 2.
    Brenner, S., Scott, L.: The mathematical theory of finite element methods. Springer, New York, 1994Google Scholar
  3. 3.
    Carr, J.C., Beatson, R.K., Cherrie, J.B., Mitchell, T.J., Fright, W.R. McCallum, B.C., Evans, T.R.: Reconstruction and representation of 3D objects with radial basis functions. In: Computer graphics proceedings, annual conference series, Addison Wesley, 2001, pp. 67–76Google Scholar
  4. 4.
    Cox, D.D.: Multivariate smoothing spline functions. SIAM J. Numer. Anal. 21, 789–813 (1984)CrossRefGoogle Scholar
  5. 5.
    Craven, P., Wahba, G.: Smoothing noisy data with spline functions: estimating the correct degree of smoothing by the method of generalized cross-validation. Numer. Math. 31, 377–403 (1979)CrossRefGoogle Scholar
  6. 6.
    Cristianini, N., Shawe-Taylor, J.: An introduction to support vector machines and other kernel-based learning methods. Cambridge University Press, Cambridge, 2000Google Scholar
  7. 7.
    Cucker, F., Smale, S.: On the mathematical foundation of learning. Bull. Amer. Math. Soc. 39, 1–49 (2001)CrossRefGoogle Scholar
  8. 8.
    de Boor, C.: A practical guide to splines. Springer, New York, revised ed., 2001Google Scholar
  9. 9.
    Duchon, J.: Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In: Constructive theory of functions of several variables, Schempp, W., Zeller, K. (eds.) Berlin, Springer, 1977, pp. 85–100,Google Scholar
  10. 10.
    Duchon, J., Sur l'erreur d'interpolation des fonctions de plusieurs variables par les Dm-splines. Rev. Française Automat. Informat. Rech. Opér. Anal. Numer. 12, 325–334 (1978)Google Scholar
  11. 11.
    Evgeniou, T., Pontil, M., Poggio, T.: Regularization networks and support vector machines. Adv. Comput. Math. 13, 1–50 (2000)CrossRefGoogle Scholar
  12. 12.
    Golitschek, M.V., Schumaker, L.L.: Data fitting by penalized least squares. In: Algorithms for approximation II, Mason, C., Cox, M.G. (eds.), London, Chapman and Hall, 1990, pp. 210–227Google Scholar
  13. 13.
    Kersey, S.N.: Near-interpolation. Numer. Math. 94 523–540 (2003)Google Scholar
  14. 14.
    Kersey, S.N., On the problem of smoothing and near-interpolation. Math. Comput. 72, 1873–1895 (2003)Google Scholar
  15. 15.
    Narcowich, F.J., Ward, J.D. (1991) Norms of inverses and condition numbers for matrices associated with scattered data. J. Approx. Theory 64, 69–94 (1991)Google Scholar
  16. 16.
    Narcowich, F.J., Norms of inverses for matrices associated with scattered data. In: Curves and surfaces, Laurent, P.-J., Méhauté, A.L., Schumaker, L.L. (eds.), Boston, Academic Press, 1991, pp. 341–348Google Scholar
  17. 17.
    Narcowich, F.J., Norm estimates for the inverse of a general class of scattered-data radial-function interpolation matrices. J. Approx. Theory, 69, 84–109 (1992)Google Scholar
  18. 18.
    Narcowich, F.J., On condition numbers associated with radial-function interpolation. J. Math. Anal. Appl. 186, 457–485 (1994)Google Scholar
  19. 19.
    Narcowich, F.J., Scattered-data interpolation on ℝn: Error estimates for radial basis and band-limited functions. SIAM J. Math. Anal. 36, 284–300 (2004)CrossRefGoogle Scholar
  20. 20.
    Narcowich, F.J., Ward, J.D., Wendland, H.: Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting. Math. Comput. 74, 643–763 (2005)Google Scholar
  21. 21.
    Ragozin, D.L.: Error bounds for derivative estimates based on spline smoothing of exact or noisy data. J. Approx. Theory 37, 335–355 (1983)CrossRefGoogle Scholar
  22. 22.
    Reinsch, C.H.: Smoothing by spline functions. Numer. Math. 10, 177–183 (1967)CrossRefGoogle Scholar
  23. 23.
    Reinsch, C.H.: Smoothing by spline functions II. Numer. Math. 16, 451–454 (1971)CrossRefGoogle Scholar
  24. 24.
    Schaback, R.: Error estimates and condition number for radial basis function interpolation. Adv. Comput. Math. 3, 251–264 (1995)Google Scholar
  25. 25.
    Schoenberg, I.J.: Spline functions and the problem of graduation. Proc. Nat. Acad. Sci. (USA) 52, 947–950 (1964)Google Scholar
  26. 26.
    Schölkopf, B., Smola, A.J.: Learning with kernels – support vector machines, regularization, optimization, and beyond. MIT Press, Cambridge, Massachusetts, 2002Google Scholar
  27. 27.
    Schultz, M.H.: Error bounds for polynomial spline interpolation. Math. Comput. 24, 507–515 (1970)Google Scholar
  28. 28.
    Schumaker, L.L.: Spline functions - basic theory. Wiley-Interscience Publication, New York, 1981Google Scholar
  29. 29.
    Wahba, G.: Smoothing noisy data by spline functions. Numer. Math. 24, 383–393 (1975)CrossRefGoogle Scholar
  30. 30.
    Wahba, G., Spline models for observational data. CBMS-NSF, Regional Conference Series in Applied Mathematics, Siam, Philadelphia, 1990Google Scholar
  31. 31.
    Wei, T., Hon, Y., Wang, Y.B.: Reconstruction of numerical derivatives from scattered noisy data. Inverse Problems 21, 657–672 (2005)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Wendland, H.: Scattered data approximation. Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK, 2005Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Institut für Numerische und Angewandte MathematikUniversität GöttingenGöttingenGermany

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