Numerische Mathematik

, Volume 107, Issue 2, pp 181–211 | Cite as

An extension of a bound for functions in Sobolev spaces, with applications to (m, s)-spline interpolation and smoothing

  • Rémi Arcangéli
  • María Cruz López de Silanes
  • Juan José TorrensEmail author


Given a function f on a bounded open subset Ω of \({\mathbb{R}}^n\) with a Lipschitz-continuous boundary, we obtain a Sobolev bound involving the values of f at finitely many points of \(\overline\Omega\) . This result improves previous ones due to Narcowich et al. (Math Comp 74, 743–763, 2005), and Wendland and Rieger (Numer Math 101, 643–662, 2005). We then apply the Sobolev bound to derive error estimates for interpolating and smoothing (m, s)-splines. In the case of smoothing, noisy data as well as exact data are considered.

Mathematics Subject Classification (2000)

41A25 41A05 41A15 


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  1. 1.
    Adams R.A. (1975). Sobolev Spaces. Academic, New York zbMATHGoogle Scholar
  2. 2.
    Adams R.A. and Fournier J.J.F. (2003). Sobolev Spaces. Academic, New York zbMATHGoogle Scholar
  3. 3.
    Arcangéli R., López de Silanes M.C. and Torrens J.J. (2004). Multidimensional Minimizing Splines. Theory and Applications. Grenoble Science. Kluwer Academic Publishers, Boston Google Scholar
  4. 4.
    Arcangéli R. and Ycart B. (1993). Almost sure convergence of smoothing D m-splines for noisy data. Numer. Math. 66: 281–294 zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bezhaev A.Y. and Vasilenko V.A. (2001). Variational Theory of Splines. Kluwer Academic/Plenum Publishers, New York zbMATHGoogle Scholar
  6. 6.
    Bouleau N. (1986). Probabilités de l’Ingénieur. Hermann, Paris zbMATHGoogle Scholar
  7. 7.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, vol. 40. SIAM, Philadelphia (2002). Firstly published by North-Holland, Amsterdam (1978)Google Scholar
  8. 8.
    Cox D. (1984). Multivariate smoothing spline functions. SIAM J. Numer. Anal. 21(4): 789–813 zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Craven P. and Wahba G. (1979). Smoothing noisy data with spline functions. Numer. Math. 31: 377–403 zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Duchon J. (1977). Splines minimizing rotation-invariant semi-norms in Sobolev spaces. Lecture Notes Math. 571: 85–100 CrossRefMathSciNetGoogle Scholar
  11. 11.
    Duchon J. (1978). Sur l’erreur d’interpolation des fonctions de plusieurs variables par les D m-splines. RAIRO Anal. Numer. 12(4): 325–334 zbMATHMathSciNetGoogle Scholar
  12. 12.
    Grisvard P. (1985). Elliptic Problems in Nonsmooth Domains. Pitman, Boston zbMATHGoogle Scholar
  13. 13.
    Johnson M.J. (2004). An error analysis for radial basis function interpolation. Numer. Math. 98: 675–694 zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Johnson M.J. (2004). The L p-approximation order of surface spline interpolation for 1 ≤ p ≤ 2. Constr. Approx. 20: 303–324 zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Le Gia Q.T., Narcowich F.J., Ward J.D. and Wendland H. (2006). Continuous and discrete least-squares approximation by radial basis functions on spheres. J. Approx. Theory 143: 124–133 zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Light W. and Wayne H. (1998). On power functions and error estimates for radial basis function interpolation. J. Approx. Theory 92: 245–266 zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    López de Silanes M.C. and Arcangéli R. (1989). Estimations de l’erreur d’approximation par splines d’interpolation et d’ajustement d’ordre (m, s). Numer. Math. 56: 449–467 zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    López de Silanes M.C. and Arcangéli R. (1991). Sur la convergence des D m-splines d’ajustement pour des données exactes ou bruitées. Rev. Mat. Univ. Complut. Madrid 4(2–3): 279–294 zbMATHMathSciNetGoogle Scholar
  19. 19.
    Madych W.R. (2006). An estimate for multivariate interpolation II. J. Approx. Theory 142: 116–128 zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Madych W.R. and Potter E.H. (1985). An estimate for multivariate interpolation. J. Approx. Theory 43: 132–139 zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Narcowich F.J., Ward J.D. and Wendland H. (2005). Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting. Math. Comp. 74: 743–763 zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Narcowich F.J., Ward J.D. and Wendland H. (2006). Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions. Constr. Approx. 24: 175–186 zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Nečas J. (1967). Les Méthodes Directes en Théorie des Équations Elliptiques. Masson, Paris Google Scholar
  24. 24.
    Ragozin D. (1983). Error bounds for derivative estimates based on spline smoothing of exact or noisy data. J. Approx. Theory 37: 335–355 zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Sanchez, A.M.: Sur l’estimation des erreurs d’approximation et d’interpolation polynomiales dans les espaces de Sobolev d’ordre non entier. Thèse de 3e cycle, Université de Pau et des Pays de l’Adour (1984)Google Scholar
  26. 26.
    Sanchez A.M. and Arcangéli R. (1984). Estimations des erreurs de meilleure approximation polynomiale et d’interpolation de Lagrange dans les espaces de Sobolev d’ordre non entier. Numer. Math. 45: 301–321 zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Stein E. (1970). Singular Integrals and Differentiability Properties of Functions. Princenton University Press, Princenton, New Jersey zbMATHGoogle Scholar
  28. 28.
    Strang G. (1972). Approximation in the Finite Element Method. Numer. Math. 19: 81–98 zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Utreras F. (1988). Convergence rates for multivariate smoothing spline functions. J. Approx. Theory 52: 1–27 zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Wahba G. (1975). Smoothing noisy data by spline functions. Numer. Math. 24: 383–393 zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Wendland H. and Rieger C. (2005). Approximate interpolation with applications to selecting smoothing parameters. Numer. Math. 101: 643–662 CrossRefMathSciNetGoogle Scholar
  32. 32.
    Wu Z.M. and Schaback R. (1993). Local error estimates for radial basis function interpolation to scattered data. IMA J. Numer. Anal. 13: 13–27 zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Rémi Arcangéli
    • 1
  • María Cruz López de Silanes
    • 2
  • Juan José Torrens
    • 3
    Email author
  1. 1.ArbasFrance
  2. 2.Departamento de Matemática Aplicada, C.P.S.Universidad de ZaragozaZaragozaSpain
  3. 3.Departamento de Ingeniería Matemática e InformáticaUniversidad Pública de NavarraPamplonaSpain

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