Skip to main content
Log in

Optimal Error of Analytic Continuation from a Finite Set with Inaccurate Data in Hilbert Spaces of Holomorphic Functions

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We consider the problem of analytic continuation with inaccurate data from a finite subset U of a domain D of C n to a point z 0D U for the functions f belonging to a bounded correctness set V in a Hilbert space H(D) of analytic functions in D. In the case when H(D) is a Hilbert space with a reproducing kernel, we find constructive formulas for calculating the optimal error, the optimal function, and the optimal linear algorithm for extrapolation to a point z 0 for functions in V whose approximate values are given on a set U. Moreover, we study the asymptotics of the optimal error in the case when the errors of initial data vanish.

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

Access this article

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. Lavrent'ev M. M., On Some Ill-Posed Problems of Mathematical Physics [in Russian], Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk (1962).

    Google Scholar 

  2. Marchuk A. G. and Osipenko K. Yu., “Best approximation of functions given with error at finitely many points,” Mat. Zametki, 17, No. 3, 359-368 (1975).

    Google Scholar 

  3. Melkman A. A. and Micchelli C. A., “Optimal estimation of linear operators in Hilbert spaces from inaccurate data,” Siam. J. Numer. Anal., 16, No. 1, 87-105 (1979).

    Google Scholar 

  4. Micchelli C. A. and Rivlin T. J., “Lectures on optimal recovery,” Lecture Notes in Math., 1129, 21-93 (1985).

    Google Scholar 

  5. Arestov V. V., “The best recovery of operators and related problems,” Trudy MIAN, 17, 3-20 (1989).

    Google Scholar 

  6. Miller K., “Least squares methods for ill-posed problems with a prescribed bound,” SIAM J. Math. Anal, 1, No. 1, 52-74 (1970).

    Google Scholar 

  7. Maergoîz L. S., “An error of optimal recovery in Hilbert spaces of entire functions,” in: Proceedings of the International Conference and Chebyshev's Readings Dedicated to the Occasion of 175-Birthday of P. L. Chebyshev [in Russian], Moscow Univ., Moscow, 1996, pp. 236-239.

    Google Scholar 

  8. Maergoîz L. S., “Extremal properties of entire functions of the Wiener class and their applications,” Dokl. Akad. Nauk, 356, No. 2, 161-165 (1997).

    Google Scholar 

  9. Maergoîz L. S., “An optimal estimate for extrapolation from a finite set in the Wiener class,” Sibirsk. Mat. Zh., 41, No. 6, 1363-1375 (2000).

    Google Scholar 

  10. Fedotov A. M., “The numerical algorithm for extrapolation of functions of the Wiener class,” Dokl. Akad. Nauk SSSR, 314, No. 2, 306-309 (1990).

    Google Scholar 

  11. Fedotov A. M., “Analytic continuation of functions from discrete sets,” J. Inverse Ill-Posed Problems, 2, No. 3, 235-252 (1994).

    Google Scholar 

  12. Fedotov A. M. and Settarov J. A., “Optimal algorithms in Hilbert space for the continuation of entire functions,” Scientific Siberian / Numerical and Data Analysis. Ser. A. Tassin, France (AMSE Transaction, 1994, V. 11), 70-75 (1994).

    Google Scholar 

  13. Anikonov Yu. E. and Uzakov M. M., “Stability estimates in multidimensional problems of analytic continuation,” in: Nonclassical Equations of Mathematical Physics [in Russian], Izdat. Inst. Mat., Novosibirsk, 1985, pp. 3-7.

    Google Scholar 

  14. Osipenko K. Yu. and Stesin M. I., “On some problems of optimal recovery of analytic and harmonic functions from inaccurate data,” Sibirsk. Mat. Zh., 34, No. 3, 144-160 (1993).

    Google Scholar 

  15. Hille E., “Introduction to general theory of reproducing kernels,” Rocky Mountain J. Math., 2, No. 3, 321-368 (1972).

    Google Scholar 

  16. Akhiezer N. I. and Glazman I. M., The Theory of Linear Operators in Hilbert Space [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  17. Khurgin Ya. I. and Yakovlev V. P., Compactly Supported Functions in Physics and Technics [in Russian], Nauka, Moscow (1971).

    Google Scholar 

  18. Gantmakher F. R., The Theory of Matrices [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  19. Halmos P. R., Finite Dimensional Vector Spaces [Russian translation], Fizmatgiz, Moscow (1963).

    Google Scholar 

  20. Buldyrev V. S. and Pavlov B. S., Linear Algebra. Functions in Several Variables [in Russian], Leningrad Univ., Leningrad (1985).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maergoiz, L.S., Fedotov, A.M. Optimal Error of Analytic Continuation from a Finite Set with Inaccurate Data in Hilbert Spaces of Holomorphic Functions. Siberian Mathematical Journal 42, 926–935 (2001). https://doi.org/10.1023/A:1011967711386

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1011967711386

Keywords

Navigation