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

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  • Nadezda Sukhorukova

    (Swinburne University of Technology
    Federation University Australia)

  • Julien Ugon

    (Federation University Australia)

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.

Suggested Citation

  • Nadezda Sukhorukova & Julien Ugon, 2016. "Chebyshev Approximation by Linear Combinations of Fixed Knot Polynomial Splines with Weighting Functions," Journal of Optimization Theory and Applications, Springer, vol. 171(2), pages 536-549, November.
    • Handle: RePEc:spr:joptap:v:171:y:2016:i:2:d:10.1007_s10957-016-0887-0
    DOI: 10.1007/s10957-016-0887-0
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    References listed on IDEAS

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    1. Zamir, Z. Roshan & Sukhorukova, N. & Amiel, H. & Ugon, A. & Philippe, C., 2015. "Convex optimisation-based methods for K-complex detection," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 947-956.
    2. Mirrlees, J. A., 1976. "Optimal tax theory : A synthesis," Journal of Public Economics, Elsevier, vol. 6(4), pages 327-358, November.
    3. Tuomala, Matti, 1990. "Optimal Income Tax and Redistribution," OUP Catalogue, Oxford University Press, number 9780198286059.
    4. Nadezda Sukhorukova, 2010. "Uniform Approximation by the Highest Defect Continuous Polynomial Splines: Necessary and Sufficient Optimality Conditions and Their Generalisations," Journal of Optimization Theory and Applications, Springer, vol. 147(2), pages 378-394, November.
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