Skip to main content
SpringerLink
Log in

On the Existence of an Element with Given Deviations from an Expanding System of Subspaces

  • Research Articles
  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We expand the class of deviation sequences for which the problem on the existence of an element of a Banach space with these deviations from a system of nested subspaces is solved positively regardless of the space and the system of subspaces. This result is used to narrow the gap between the weak asymptotics constants in S. V. Konyagin’s theorem on the existence of an element whose deviations are asymptotically close to given ones.

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

References

  1. S. N. Bernstein, “On an inverse problem of the theory of best approximation of continuous functions,” in Collection of Works (Izd. Akad. Nauk SSSR, Moscow, 1954), Vol. 2, pp. 292–294 [in Russian].

    Google Scholar 

  2. A. F. Timan, Theory of Approximation of Functions of a Real Variable. (Fizmatgiz, Moscow, 1960) [in Russian].

    Google Scholar 

  3. V. N. Nikol’skii, “Some properties of reflexive spaces,” Kalinin. Gos. Ped. Inst. Ucen. Zap. 29, 121–125 (1963).

    MathSciNet  Google Scholar 

  4. I. S. Tyuremskikh, “The \((B)\)-property of Hilbert spaces,” Kalinin. Gos. Ped. Inst. Ucen. Zap. 39, 53–64 (1964).

    MathSciNet  Google Scholar 

  5. H. S. Shapiro, “Some negative theorems of approximation theory,” Michigan Math. J. 11 (3), 211–217 (1964).

    Article  MathSciNet  CAS  Google Scholar 

  6. I. S. Tyuremskikh, “A certain theorem of S. N. Bernstein,” Kalinin. Gos. Ped. Inst. Ucen. Zap. 52, 123–129 (1967).

    MathSciNet  Google Scholar 

  7. P. A. Borodin, “On the existence of an element with given deviations from an expanding system of subspaces,” Math. Notes 80 (5), 621–630 (2006).

    Article  MathSciNet  Google Scholar 

  8. S. V. Konyagin, “Deviation of elements of a Banach space from a system of subspaces,” Proc. Steklov Inst. Math. 284, 204–207 (2014).

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author thanks P. A. Borodin for valuable remarks.

Funding

This work was financially supported by the Russian Science Foundation (project 23-71-30001, https://rscf.ru/en/project/23-71-30001/) at Lomonosov Moscow State University.

Author information

Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    Yu. A. Skvortsov

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991, Russia

    Yu. A. Skvortsov

Authors
  1. Yu. A. Skvortsov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu. A. Skvortsov.

Ethics declarations

The author of this work declares that he has no conflicts of interest.

Additional information

Translated from Matematicheskie Zametki, 2023, Vol. 114, pp. 780–788 https://doi.org/10.4213/mzm13982.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Skvortsov, Y.A. On the Existence of an Element with Given Deviations from an Expanding System of Subspaces. Math Notes 114, 949–956 (2023). https://doi.org/10.1134/S0001434623110299

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434623110299

Keywords

Search

Navigation