Journal of Optimization Theory and Applications

, Volume 147, Issue 2, pp 378–394 | Cite as

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

  • Nadezda Sukhorukova


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.


Nonsmooth optimisation Quasidifferentials Chebyshev approximation Polynomial splines with fixed knots 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Caratheodory, C.: Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen. Rend. Circ. Mat. Palermo 32, 193–217 (1911) MATHCrossRefGoogle Scholar
  2. 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) Google Scholar
  3. 3.
    Davydov, O.V.: A class of weak Chebyshev spaces and characterization of best approximations. J. Approx. Theory 81, 250–259 (1995) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Demyanov, V.F., Rubinov, A.M. (eds.): Quasidifferentiability and Related Topics. Kluwer, Dordrecht (2000), 400 p. MATHGoogle Scholar
  5. 5.
    Nürnberger, G.: Approximation by Spline Functions. Springer, Berlin (1989) MATHGoogle Scholar
  6. 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. 7.
    Rice, J.: Characterization of Chebyshev approximation by splines. SIAM J. Numer. Anal. 4(4), 557–567 (1967) MATHCrossRefMathSciNetGoogle Scholar
  8. 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 Google Scholar
  9. 9.
    Tarashnin, M.G.: Application of the theory of quasidifferentials to solving approximation problems. PhD thesis, St-Petersburg State University (1996), 119 p. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Center for Informatics and Applied OptimizationUniversity of BallaratBallaratAustralia
  2. 2.CIAO, ITMSUniversity of BallaratBallaratAustralia

Personalised recommendations