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Constrained L2-Approximation by Polynomials on Subsets of the Circle

  • Laurent Baratchart
  • Juliette Leblond
  • Fabien Seyfert
Chapter
Part of the Fields Institute Communications book series (FIC, volume 81)

Abstract

We study best approximation to a given function, in the least square sense on a subset of the unit circle, by polynomials of given degree which are pointwise bounded on the complementary subset. We show that the solution to this problem, as the degree goes large, converges to the solution of a bounded extremal problem for analytic functions which is instrumental in system identification. We provide a numerical example on real data from a hyperfrequency filter.

2010 Mathematics Subject Classification.

30E10 30E25 41A10 46N10 47A52 93B30 

References

  1. 1.
    L. Baratchart, J. Grimm, J. Leblond, M. Olivi, F. Seyfert, and F. Wielonsky. Identification d’un filtre hyperfréquences par approximation dans le domaine complexe, 1998. INRIA technical report no. 0219.Google Scholar
  2. 2.
    L. Baratchart, J. Grimm, J. Leblond, and J.R. Partington. Asymptotic estimates for interpolation and constrained approximation in H 2 by diagonalization of toeplitz operators. Integral equations and operator theory, 45:269–299, 2003.Google Scholar
  3. 3.
    L. Baratchart and J. Leblond. Hardy approximation to L p functions on subsets of the circle with 1 ≤ p < . Constructive Approximation, 14:41–56, 1998.Google Scholar
  4. 4.
    L. Baratchart, J. Leblond, and J.R. Partington. Hardy approximation to L functions on subsets of the circle. Constructive Approximation, 12:423–436, 1996.Google Scholar
  5. 5.
    L. Baratchart, J. Leblond, and J.R. Partington. Problems of Adamjan–Arov–Krein type on subsets of the circle and minimal norm extensions. Constructive Approximation, 16:333–357, 2000.Google Scholar
  6. 6.
    Laurent Baratchart, Sylvain Chevillard, and Fabien Seyfert. On transfer functions realizable with active electronic components. Technical Report RR-8659, INRIA, Sophia Antipolis, 2014. 36 pages.Google Scholar
  7. 7.
    J.M. Borwein and A.S. Lewis. Convex Analysis and Nonlinear Optimization. CMS Books in Math. Can. Math. Soc., 2006.Google Scholar
  8. 8.
    E. W. Cheney. Introduction to approximation theory. Chelsea, 1982.Google Scholar
  9. 9.
    J. C. Doyle, B. A. Francis, and A. R. Tannenbaum. Feedback Control Theory. Macmillan Publishing Company, 1992.Google Scholar
  10. 10.
    P.L. Duren. Theory of H p spaces. Academic Press, 1970.Google Scholar
  11. 11.
    J.B. Garnett. Bounded analytic functions. Academic Press, 1981.Google Scholar
  12. 12.
    M.G. Krein and P.Y. Nudel’man. Approximation of L 2(ω 1, ω 2) functions by minimum– energy transfer functions of linear systems. Problemy Peredachi Informatsii, 11(2):37–60, 1975. English translation.Google Scholar
  13. 13.
    J. Leblond and J. R. Partington. Constrained approximation and interpolation in Hilbert function spaces. J. Math. Anal. Appl., 234(2):500–513, 1999.Google Scholar
  14. 14.
    Martine Olivi, Fabien Seyfert, and Jean-Paul Marmorat. Identification of microwave filters by analytic and rational h 2 approximation. Automatica, 49(2):317–325, 2013.Google Scholar
  15. 15.
    Jonathan Partington. Linear operators and linear systems. Number 60 in Student texts. London Math. Soc., 2004.Google Scholar
  16. 16.
    Rik Pintelon, Yves Rollain, and Johan Schoukens. System Identification: A Frequency Domain Approach. Wiley, 2012.Google Scholar
  17. 17.
    W. Rudin. Real and complex analysis. McGraw–Hill, 1987.Google Scholar
  18. 18.
    A. Schneck. Constrained optimization in hardy spaces. Preprint, 2009.Google Scholar
  19. 19.
    F. Seyfert. Problèmes extrémaux dans les espaces de Hardy. These de Doctorat, Ecole des Mines de Paris, 1998.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Laurent Baratchart
    • 1
  • Juliette Leblond
    • 1
  • Fabien Seyfert
    • 1
  1. 1.INRIASophia-Antipolis CedexFrance

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