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Best L 1 approximation of Heaviside-type functions from Chebyshev and weak-Chebyshev spaces

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Abstract

In this article, we study the problem of best L 1 approximation of Heaviside-type functions from Chebyshev and weak-Chebyshev spaces. We extend the Hobby-Rice theorem (Proc. Am. Math. Soc., 16, 665–670, 1965) into an appropriate framework and prove the unicity of best L 1 approximation of Heaviside-type functions from an even-dimensional Chebyshev space under some assumptions on the dimension of the subspaces composed of the odd and even functions. We also apply the results to compute best L 1 approximations of Heaviside-type functions by polynomials and Hermite polynomial splines with fixed knots.

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Authors and Affiliations

  1. Arts et Métiers ParisTech, LSIS - UMR CNRS 7296, 8 Boulevard Louis XIV, 59046, Lille Cedex, France

    Laurent Gajny, Olivier Gibaru & Eric Nyiri

  2. INRIA Lille-Nord-Europe, NON-A Research Team, 40, Avenue Halley, 59650, Villeneuve d’Ascq, France

    Olivier Gibaru

  3. Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, 27695, USA

    Shu-Cherng Fang

Authors
  1. Laurent Gajny
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  2. Olivier Gibaru
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  3. Eric Nyiri
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  4. Shu-Cherng Fang
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Correspondence to Laurent Gajny.

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Gajny, L., Gibaru, O., Nyiri, E. et al. Best L 1 approximation of Heaviside-type functions from Chebyshev and weak-Chebyshev spaces. Numer Algor 75, 827–843 (2017). https://doi.org/10.1007/s11075-016-0222-8

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  • DOI: https://doi.org/10.1007/s11075-016-0222-8

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