Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Optimal segmented approximations

Optimale segmentierte Approximationen

  • 19 Accesses

  • 12 Citations


Segmented approximations with free knots for a given continuous function are discussed in a general form. This form includes any kind of continuous approximating segments, and a great variety of error criteria, such as anyL s -norm, or the maximum norm of the pointwise error. It is shown that a solution to the problem exists under very general conditions, and that one solution at least must have continuous pointwise error modulus. Finally, algorithms for the iterative reduction of the approximation error are proposed, which lead to good, but not necessarily best approximations.


Segmentierte Approximationen mit freien Knoten für eine gegebene stetige Funktion werden in einer allgemeinen Form diskutiert. Diese Form umfaßt jede Art von stetigen approximierenden Segmenten und eine große Vielfalt von Fehlerkriterien, z. B. irgend eineL s -Norm, oder die Maximum-Norm des punktweisen Fehlers. Es wird gezeigt, daß die Existenz einer Lösung unter sehr allgemeinen Bedingungen gesichert ist und daß mindestens eine Lösung stetigen Betrag des punktweisen Fehlers haben muß. Schließlich werden Algorithmen für die iterative Reduktion des Approximationsfehlers vorgeschlagen. Diese Algorithmen liefern gute, aber nicht unbedingt beste Approximationen.

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


  1. [1]

    Lawson, C. L.: Characteristic properties of the segmented rational minimax approximation problem. Numer. Math.6, 293–301 (1964).

  2. [2]

    Phillips, G. M.: Error estimates for best polynomial approximations, in: Approximation Theory (Talbot, A., ed.), pp. 1–5. New York: Academic Press 1970.

  3. [3]

    Phillips, G. M.: Best polynomial approximations, a corrected proof. BIT19, 95–97 (1979).

  4. [4]

    Cox, M. G.: Curve fitting with piecewise polynomials. J. Inst. Math. Appl.8, 36–52 (1971).

  5. [5]

    Pavlidis, T., Maika, A. P.: Uniform piecewise polynomial approximation with variable joints. J. Approx. Th.12, 61–69 (1974).

  6. [6]

    Schumaker, L. L.:L 2 approximation by splines with free knots. (To appear in the proceedings of a conference in Siegen in spring 1979.)

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kioustelidis, J.B. Optimal segmented approximations. Computing 24, 1–8 (1980). https://doi.org/10.1007/BF02242787

Download citation


  • Continuous Function
  • Approximation Error
  • General Condition
  • Computational Mathematic
  • Maximum Norm