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

• Published in: Journal of optimization theory and applications Vol. 171; no. 2; pp. 536 - 549
• Main Authors: Sukhorukova, Nadezda, Ugon, Julien
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.11.2016; Springer Nature B.V
• Subjects: 20th century; Applications of Mathematics; Approximation; Calculus of Variations and Optimal Control; Optimization; Deviation; Engineering; Knots; Linear programming; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Optimization techniques; Polynomials; Signal processing; Splines; Studies; Theory of Computation; Alternance; 90C51; Polynomial splines; Uniform approximation; 41A50; 41A15; Weighting functions
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-016-0887-0

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.