Skip to main content
SpringerLink
Log in

Optimal Recovery of a Derivative of an Analytic Function from Values of the Function Given with an Error on a Part of the Boundary. II

  • Published:
Analysis Mathematica Aims and scope Submit manuscript

Abstract

We continue the study of several related extremal problems for functions analytic in a simply connected domain G with a rectifiable Jordan boundary Γ. In particular, the problem of optimal recovery of a derivative at a point z0 ∈ G from limit boundary values given with an error on a measurable part γ1 of the boundary Γ for the class Q of functions with limit boundary values bounded by 1 on γ0 = Γ γ1 as well as the problem of the best approximation of the derivative at a point z0 ∈ G by bounded linear functionals in L∞(γ1) on the class Q. Complete exact solutions of the considered problems are obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. R. Akopyan, Optimal recovery of a derivative of an analytic function from values of the function given with an error on a part of the boundary, Analysis Math., 44 (2018), 3–19.

    Article  MathSciNet  Google Scholar 

  2. R. R. Akopyan, Approximation of derivatives of analytic functions from one Hardy class by another Hardy class, Trudy Inst. Mat. Mekh. UrO RAN, 25 (2019), 21–29 (in Russian).

    Article  Google Scholar 

  3. R. R. Akopyan, An analogue of the two-constants theorem and optimal recovery of analytic functions, Mat. Sb., 210 (2019), 3–16 (in Russian); translation in Sb. Math., 210 (2019), 1348–1360.

    Article  MathSciNet  Google Scholar 

  4. V. V. Arestov, Best approximation of a differentiation operator on the set of smooth functions with exactly or approximately given fourier transform, in: Mathematical Optimization Theory and Operations Research (MOTOR 2019), Lecture Notes in Computer Science, vol. 11548, Springer (Cham, 2019), 434–448.

    Chapter  Google Scholar 

  5. V. V. Arestov, Best uniform approximation of the differentiation operator by operators bounded in the space L2, Trudy Inst. Mat. Mekh. UrO RAN, 24 (2018), 34–56 (in Russian).

    Article  Google Scholar 

  6. V. V. Arestov, Approximation of unbounded operators by bounded ones and related extremal problems, Uspekhi Mat. Nauk, 51 (1996), 89–124 (in Russian); translation in Russ. Math. Surveys, 51 (1996), 1093–1126.

    Article  MathSciNet  Google Scholar 

  7. A. V. Arutyunov and K. Yu. Osipenko, Recovering linear operators and Lagrange function minimality condition, Sibirsk. Mat. Zh., 59 (2018), 15–28 (in Russian); translation in Siberian Math. J., 59 (2018), 11–21.

    MathSciNet  MATH  Google Scholar 

  8. L. Baratchart, J. Leblond, and F. Seyfert, Constrained extremal problems in H2 and Carleman’s formulae, Sb. Math., 209 (2018), 922–957 (in Russian); translation in Sb. Math., 209 (2018), 922–957.

    Article  MathSciNet  Google Scholar 

  9. V. N. Gabushin, Best approximations of functionals on certain sets, Mat. Zametki, 8 (1970), 551–562 (in Russian); translation in Math. Notes, 8 (1970), 780–785.

    MathSciNet  Google Scholar 

  10. G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Gostekhizdat (Moscow, Leningrad, 1952) (in Russian); translation: American Mathematical Society (Providence, RI, 1969).

    MATH  Google Scholar 

  11. G. G. Magaril-Il’yaev and K. Yu. Osipenko, Optimal recovery of functionals based on inaccurate data, Mat. Zametki, 50 (1991), 85–93 (in Russian); translation in Math. Notes, 50 (1991), 1274–1279.

    MATH  Google Scholar 

  12. K. Yu. Osipenko, Optimal Recovery of Analytic Functions, Nova Science (Huntington, NY, 2000).

    Google Scholar 

  13. K. Yu. Osipenko, Recovery of derivatives for functions defined on the semiaxis, J. Complexity, 48 (2018), 111–118.

    Article  MathSciNet  Google Scholar 

  14. K. Yu. Osipenko, Generalized adaptive versus nonadaptive recovery from noisy information, J. Complexity, 53 (2019), 162–172.

    Article  MathSciNet  Google Scholar 

  15. I. I. Privalov, Boundary Properties of Analytic Functions, Gosudarstv. Izdat. Tekhn.-Teor. Lit. (Moscow, Leningrad, 1950) (in Russian).

    Google Scholar 

  16. S. B. Stechkin, Best approximation of linear operators, Mat. Zametki, 1 (1967), 137–148 (in Russian); translation in Math. Notes, 1 (1967), 91–99.

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgement

The author is grateful to Professor V.V. Arestov for the attention to the study and useful discussions of the results.

Author information

Authors and Affiliations

  1. Institute of Natural Sciences and Mathematics, Ural Federal University, pr. Lenina 51, Yekaterinburg, 620000, Russia

    R. R. Akopyan

  2. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, S. Kovalevskaya Str. 16, Yekaterinburg, 620990, Russia

    R. R. Akopyan

Authors
  1. R. R. Akopyan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. R. Akopyan.

Additional information

This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00336), and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University), and as part of research conducted in the Ural Mathematical Center.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Akopyan, R.R. Optimal Recovery of a Derivative of an Analytic Function from Values of the Function Given with an Error on a Part of the Boundary. II. Anal Math 46, 409–424 (2020). https://doi.org/10.1007/s10476-020-0039-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10476-020-0039-5

Key words and phrases

Mathematics Subject Classification

Search

Navigation