Characterization theorem for best polynomial spline approximation with free knots, variable degree and fixed tails
- Authors: Crouzeix, Jean-Pierre , Sukhorukova, Nadezda , Ugon, Julien
- Date: 2017
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 172, no. 3 (2017), p. 950-964
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- Description: In this paper, we derive a necessary condition for a best approximation by piecewise polynomial functions of varying degree from one interval to another. Based on these results, we obtain a characterization theorem for the polynomial splines with fixed tails, that is the value of the spline is fixed in one or more knots (external or internal). We apply nonsmooth nonconvex analysis to obtain this result, which is also a necessary and sufficient condition for inf-stationarity in the sense of Demyanov-Rubinov. This paper is an extension of a paper where similar conditions were obtained for free tails splines. The main results of this paper are essential for the development of a Remez-type algorithm for free knot spline approximation.
Chebyshev multivariate polynomial approximation and point reduction procedure
- Authors: Sukhorukova, Nadezda , Ugon, Julien , Yost, David
- Date: 2021
- Type: Text , Journal article
- Relation: Constructive Approximation Vol. 53, no. 3 (2021), p. 529-544
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: We apply the methods of nonsmooth and convex analysis to extend the study of Chebyshev (uniform) approximation for univariate polynomial functions to the case of general multivariate functions (not just polynomials). First of all, we give new necessary and sufficient optimality conditions for multivariate approximation, and a geometrical interpretation of them which reduces to the classical alternating sequence condition in the univariate case. Then, we present a procedure for verification of necessary and sufficient optimality conditions that is based on our generalization of the notion of alternating sequence to the case of multivariate polynomials. Finally, we develop an algorithm for fast verification of necessary optimality conditions in the multivariate polynomial case. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.