Skip to main content
SpringerLink
Log in

Uniform Approximation by the Highest Defect Continuous Polynomial Splines: Necessary and Sufficient Optimality Conditions and Their Generalisations

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper necessary and sufficient optimality conditions for uniform approximation of continuous functions by polynomial splines with fixed knots are derived. The obtained results are generalisations of the existing results obtained for polynomial approximation and polynomial spline approximation. The main result is two-fold. First, the generalisation of the existing results to the case when the degree of the polynomials, which compose polynomial splines, can vary from one subinterval to another. Second, the construction of necessary and sufficient optimality conditions for polynomial spline approximation with fixed values of the splines at one or both borders of the corresponding approximation interval.

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. Caratheodory, C.: Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen. Rend. Circ. Mat. Palermo 32, 193–217 (1911)

    Article  MATH  Google Scholar 

  2. Chebyshev, P.L.: The theory of mechanisms known as parallelograms. Selected works, Publishing House of the USSR Academy of Sciences, Moscow (1955), pp. 611–648 (Russian)

  3. Davydov, O.V.: A class of weak Chebyshev spaces and characterization of best approximations. J. Approx. Theory 81, 250–259 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Demyanov, V.F., Rubinov, A.M. (eds.): Quasidifferentiability and Related Topics. Kluwer, Dordrecht (2000), 400 p.

    MATH  Google Scholar 

  5. Nürnberger, G.: Approximation by Spline Functions. Springer, Berlin (1989)

    MATH  Google Scholar 

  6. Remez, E.Ya.: General Computational Methods of Chebyshev Approximation. Problems with Linear Real Parameters. Izv. Akad. Nauk Ukrain. SSR, Kiev (1957). Translation available from United States Atomic Commission, Washington, D.C., 1962

    Google Scholar 

  7. Rice, J.: Characterization of Chebyshev approximation by splines. SIAM J. Numer. Anal. 4(4), 557–567 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  8. Soukhoroukova, N.V.: A generalisation of Valle-Poussin theorem and exchange basis rules to the case of polynomial splines. In: Proceedings of Stability and Control Processes SCP2005, dedicated to the 75-years anniversary of V.I. Zubov, vol. 2, pp. 948–958

  9. Tarashnin, M.G.: Application of the theory of quasidifferentials to solving approximation problems. PhD thesis, St-Petersburg State University (1996), 119 p.

Download references

Author information

Authors and Affiliations

  1. Center for Informatics and Applied Optimization, University of Ballarat, Ballarat, Victoria, 3353, Australia

    Nadezda Sukhorukova

  2. CIAO, ITMS, University of Ballarat, PO Box 663, Ballarat, Victoria, 3353, Australia

    Nadezda Sukhorukova

Authors
  1. Nadezda Sukhorukova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nadezda Sukhorukova.

Additional information

Communicated by V.F. Dem’yanov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sukhorukova, N. Uniform Approximation by the Highest Defect Continuous Polynomial Splines: Necessary and Sufficient Optimality Conditions and Their Generalisations. J Optim Theory Appl 147, 378–394 (2010). https://doi.org/10.1007/s10957-010-9715-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-010-9715-0

Keywords

Search

Navigation