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Finite alternation theorems and a constructive approach to piecewise polynomial approximation in chebyshev norm

- Crouzeix, Jean-Pierre, Sukhorukova, Nadezda, Ugon, Julien

  • Authors: Crouzeix, Jean-Pierre , Sukhorukova, Nadezda , Ugon, Julien
  • Date: 2020
  • Type: Text , Journal article
  • Relation: Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 123-147. http://purl.org/au-research/grants/arc/DP180100602
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  • Description: One of the purposes in this paper is to provide a better understanding of the alternance property which occurs in Chebyshev polynomial approximation and continuous piecewise polynomial approximation problems. In the first part of this paper, we prove that alternating sequences of any continuous function are finite in any given segment and then propose an original approach to obtain new proofs of the well known necessary and sufficient optimality conditions. There are two main advantages of this approach. First of all, the proofs are intuitive and easy to understand. Second, these proofs are constructive and therefore they lead to new alternation-based algorithms. In the second part of this paper, we develop new local optimality conditions for free knot polynomial spline approximation. The proofs for free knot approximation are relying on the techniques developed in the first part of this paper. The piecewise polynomials are required to be continuous on the approximation segment. © 2020, Springer Nature B.V.

Characterization theorem for best polynomial spline approximation with free knots, variable degree and fixed tails

- Crouzeix, Jean-Pierre, Sukhorukova, Nadezda, Ugon, Julien

  • 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.
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