Chebyshev Approximation by Linear Combinations of Fixed Knot Polynomial Splines with Weighting Functions
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 l...
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| Veröffentlicht in: | Journal of optimization theory and applications 2016-11, Vol.171 (2), p.536-549 |
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| creator | Sukhorukova, Nadezda Ugon, Julien |
| description | 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. |
| doi_str_mv | 10.1007/s10957-016-0887-0 |
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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). 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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.</description><subject>20th century</subject><subject>Applications of Mathematics</subject><subject>Approximation</subject><subject>Calculus of Variations and Optimal Control; Optimization</subject><subject>Deviation</subject><subject>Engineering</subject><subject>Knots</subject><subject>Linear programming</subject><subject>Mathematical analysis</subject><subject>Mathematical models</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Operations Research/Decision Theory</subject><subject>Optimization</subject><subject>Optimization techniques</subject><subject>Polynomials</subject><subject>Signal processing</subject><subject>Splines</subject><subject>Studies</subject><subject>Theory of 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Approximation by Linear Combinations of Fixed Knot Polynomial Splines with Weighting Functions</title><author>Sukhorukova, Nadezda ; Ugon, Julien</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c301t-53ccbf4de6fe045d737de75b9bca67c91412fca66138961623789b14724c652a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>20th century</topic><topic>Applications of Mathematics</topic><topic>Approximation</topic><topic>Calculus of Variations and Optimal Control; Optimization</topic><topic>Deviation</topic><topic>Engineering</topic><topic>Knots</topic><topic>Linear programming</topic><topic>Mathematical analysis</topic><topic>Mathematical models</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Operations Research/Decision Theory</topic><topic>Optimization</topic><topic>Optimization techniques</topic><topic>Polynomials</topic><topic>Signal 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| 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 |
| title | Chebyshev Approximation by Linear Combinations of Fixed Knot Polynomial Splines with Weighting Functions |
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