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

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Abstract

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.

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Authors and Affiliations

  1. Swinburne University of Technology, Melbourne, Australia

    Nadezda Sukhorukova

  2. Federation University Australia, Ballarat, Australia

    Nadezda Sukhorukova

  3. Centre for Informatics and Applied Optimization (CIAO), Federation University Australia, Ballarat, Australia

    Julien Ugon

Authors
  1. Nadezda Sukhorukova
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  2. Julien Ugon
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Correspondence to Nadezda Sukhorukova.

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Sukhorukova, N., Ugon, J. Chebyshev Approximation by Linear Combinations of Fixed Knot Polynomial Splines with Weighting Functions. J Optim Theory Appl 171, 536–549 (2016). https://doi.org/10.1007/s10957-016-0887-0

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  • DOI: https://doi.org/10.1007/s10957-016-0887-0

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