Skip to main content
SpringerLink
Log in

Natural spline functions, their associated eigenvalue problem

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

We consider the problem of studying the behaviour of the eigenvalues associated with spline functions with equally spaced knots. We show that they are\(O\left( {\frac{{i^{2m} }}{n}} \right)i = 1, \ldots ,n - m\) wherem is the order of the spline andn, the number of knots.

This result is of particular interest to prove optimality properties of the Generalized Cross-Validation Method and had been conjectured by Craven and Wahba in a recent paper.

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. Ahlberg, J., Nilson, E., Walsh, J.: The Theory of Splines and Their Applications. New York: Academic Press, 1967

    Google Scholar 

  2. Atteia, M.: Théorie et Applications de Fonctions Spline en Analyse Numérique. Thesis. Grenoble, 1966

  3. Bellman, R.: Introduction to Matrix Analysis. New York: McGraw Hill, 1960

    Google Scholar 

  4. Craven, P., Wahba, G.: Smoothing Noisy Data with Spline Function. Estimating the Correct Degree of Smoothing by the Method of Generalized Cross-Validation. Numer. Math.31, 377–403 (1979)

    Google Scholar 

  5. Duc-Jacquet, M.: Résolution de Quelques Problemes d'Analyse Numérique a l'aide de Perturbations par des Matrices Antyscalaires. Thesis. Grenoble,

  6. Gantmacher, M.: Matrix Theory. Tome II. Chelsea Publ. Co., New York, 1960

    Google Scholar 

  7. Golub, G., Heath, M., Wahba, G.: Generalized Cross-Validation as a Method for choosing a Good Ridge Parameter. Technometrics21, 2, 215–223 (1979)

    Google Scholar 

  8. Kac, M., Murdock, W.L., Szego, G.: On the Eigenvalues of Certain Hermitian forms. Canadian J. Math.

  9. Laurent, P.J.: Approximation et Optimisation. Paris: Hermman, 1972

    Google Scholar 

  10. Reinsch, C.: Oscillation Matrices with Spline Smoothing. Numer. Math.24, 375–382 (1975)

    Google Scholar 

  11. Schoenberg, I.J.: Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions. Quaterly Appl. Math.4, 1, 2 (1946)

    Google Scholar 

  12. Schumaker, L.: Spline Functions: Basic Theory. New York: John Wiley and Sons, 1981

    Google Scholar 

  13. Utreras, F.: Sur le Choix du Parametre d'Ajustement dans le Lissage par Fonctions Spline. Numer. Math.34, 1, 15–28 (1980)

    Google Scholar 

  14. Utreras, F.: Cross-Validation Techniques for Smoothing Spline Functions in One or Two Dimensions. In: Smoothing Techniques for Curve Estimation. Gasser, T., Rosemblatt, M. (eds.). Lecture Notes in Mathematics, No. 757, 1979

  15. Wahba, G.: Practical Approximate Solutions to Linear Operator Equations when the Data are Noisy. SIAM J. Numer. Anal.14, 4, 651–667 (1977)

    Google Scholar 

  16. Wahba, G.: Smoothing Noisy Data with Spline Functions. Numer. Math.24, 383–393 (1975)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matematicas, Facultad de Ciencias Fisicas y Matematicas, Universidad de Chile, Casilla 5272, Correo 3, Santiago, Chile

    Florencio Utreras

Authors
  1. Florencio Utreras
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Utreras, F. Natural spline functions, their associated eigenvalue problem. Numer. Math. 42, 107–117 (1983). https://doi.org/10.1007/BF01400921

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01400921

Subject Classifications

Search

Navigation