Siberian Mathematical Journal

, Volume 59, Issue 1, pp 11–21 | Cite as

Recovering Linear Operators and Lagrange Function Minimality Condition

  • A. V. ArutyunovEmail author
  • K. Yu. Osipenko


This article concerns the recovery of the operators by noisy information in the case that their norms are defined by integrals over infinite intervals. We study the conditions under which the dual extremal problem (often nonconvex) can be solved using the Lagrange function minimality condition.


optimal recovery linear operator extremal problem Lagrange function 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Smolyak S. A., On Optimal Recovery of Functions and Functionals over Them [Russian], Diss. Kand. Fiz.-Mat. Nauk, Moscow Univ., Moscow (1965).Google Scholar
  2. 2.
    Marchuk A. G. and Osipenko K. Yu., “Best approximation of functions specified with an error at a finite number of points,” Math. Notes, vol. 17, no. 3, 207–212 (1975).CrossRefzbMATHGoogle Scholar
  3. 3.
    Osipenko K. Yu., “Best approximation of analytic functions from information about their values at a finite number of points,” Math. Notes, vol. 19, no. 1, 17–23 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Micchelli C. A. and Rivlin T. J., “A survey of optimal recovery,” in: Optimal Estimation in Approximation Theory (C. A. Micchelli and T. J. Rivlin; eds.), Plenum Press, New York, 1977, 1–54.CrossRefGoogle Scholar
  5. 5.
    Arestov V. V., “Optimal recovery of operators, and related problems,” Trudy Mat. Inst. Steklov., vol. 189, 3–20 (1989).MathSciNetGoogle Scholar
  6. 6.
    Traub J. F. and Woźniakowski H., A General Theory of Optimal Algorithms, Academic Press, New York (1980).zbMATHGoogle Scholar
  7. 7.
    Plaskota L., Noisy Information and Computational Complexity, Cambridge Univ. Press, Cambridge (1996).CrossRefzbMATHGoogle Scholar
  8. 8.
    Osipenko K. Yu., Optimal Recovery of Analytic Functions, Nova Sci. Publ., Inc., Huntington and New York (2000).Google Scholar
  9. 9.
    Melkman A. A. and Micchelli C. A., “Optimal estimation of linear operators in Hilbert spaces from inaccurate data,” SIAM J. Numer. Anal., vol. 16, 87–105 (1979).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Magaril-Il’yaev G. G. and Osipenko K. Yu., “Optimal recovery of functions and their derivatives from Fourier coefficients prescribed with an error,” Sb. Math., vol. 193, no. 3, 387–407 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Osipenko K. Yu., “The Hardy–Littlewood–Pólya inequality for analytic functions in Hardy–Sobolev spaces,” Sb. Math., vol. 197, no. 3, 315–334 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Magaril-Il’yaev G. G. and Osipenko K. Yu., “Optimal recovery of the solution of the heat equation from inaccurate data,” Sb. Math., vol. 200, no. 5, 665–682 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Arutyunov A. V., “Milyutin’s theorem in linear-quadratic optimal control problems,” Differ. Equ., vol. 37, no. 11, 1627–1630 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Arutyunov A. V., “Lagrange principle in quadratic optimal control problems with infinite horizon,” Differ. Equ., vol. 45, no. 11, 1595–1601 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ioffe A. D. and Tikhomirov V. M., “Some remarks on variational principles,” Math. Notes, vol. 61, no. 2, 248–253 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bobylev N. A., Emel’yanov S. V., and Korovin S. K., Geometric Methods in Variational Problems [Russian], Magistr, Moscow (1998).zbMATHGoogle Scholar
  17. 17.
    Arutyunov A. V., “Smooth abnormal problems in extremum theory and analysis,” Russian Math. Surveys, vol. 67, no. 3 (405), 403–457 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Arutyunov A. V., “Approximation to solutions of linear control systems by compactly supported solutions,” Differ. Equ., vol. 51, no. 6, 792–797 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Yosida K., Functional Analysis, Springer-Verlag, Berlin (1994).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Peoples’ Friendship University of RussiaMoscowRussia
  2. 2.Moscow State University Institute for Information Transmission ProblemsMoscowRussia

Personalised recommendations