Journal of Optimization Theory and Applications

, Volume 171, Issue 2, pp 536–549 | Cite as

Chebyshev Approximation by Linear Combinations of Fixed Knot Polynomial Splines with Weighting Functions

  • Nadezda Sukhorukova
  • Julien Ugon


In this paper, we derive conditions for best uniform approximation by fixed knots polynomial splines with weighting functions. The theory of Chebyshev approximation for fixed knots polynomial functions is very elegant and complete. Necessary and sufficient optimality conditions have been developed leading to efficient algorithms for constructing optimal spline approximations. The optimality conditions are based on the notion of alternance (maximal deviation points with alternating deviation signs). In this paper, we extend these results to the case when the model function is a product of fixed knots polynomial splines (whose parameters are subject to optimization) and other functions (whose parameters are predefined). This problem is nonsmooth, and therefore, we make use of convex and nonsmooth analysis to solve it.


Uniform approximation Polynomial splines Alternance  Weighting functions 

Mathematics Subject Classification

90C51 41A15 41A50 


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Swinburne University of TechnologyMelbourneAustralia
  2. 2.Federation University AustraliaBallaratAustralia
  3. 3.Centre for Informatics and Applied Optimization (CIAO)Federation University AustraliaBallaratAustralia

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