Skip to main content
SpringerLink
Log in

On the best approximations and rate of convergence of decompositions in the root vectors of an operator

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We establish upper bounds of the best approximations of elements of a Banach space B by the root vectors of an operator A that acts in B. The corresponding estimates of the best approximations are expressed in terms of a K-functional associated with the operator A. For the operator of differentiation with periodic boundary conditions, these estimates coincide with the classical Jackson inequalities for the best approximations of functions by trigonometric polynomials. In terms of K-functionals, we also prove the abstract Dini-Lipschitz criterion of convergence of partial sums of the decomposition of f from B in the root vectors of the operator A to f

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. D. Jackson, Über die Genauigkeit der Annäherung stetiger Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrische Summer gegebener Ordnung: Inauguraldisertation und Preischrift, University of Göttingen (1911).

  2. S. B. Stechkin, “On the order of the best approximations of continuous functions,” Izv. Akad. Nauk SSSR, Ser. Mat., 15, No. 3, 219–242 (1951).

    MATH  Google Scholar 

  3. N. I. Akhiezer, Lectures on the Theory of Approximations [in Russian], Gostekhteorizdat, Moscow-Leningrad (1947).

    Google Scholar 

  4. G. V. Radzievskii, “The rate of convergence of decompositions of ordinary functional-differential operators by eigenfunctions,” in: Some Problems of the modern Theory of Differential equations, Preprint No. 94.29, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1994), pp. 14–27.

    Google Scholar 

  5. G. V. Radzievskii, “Boundary-value problems and related moduli of continuity,” Funkts. Anal. Prilozh., 29, No. 3, 87–90 (1995).

    MathSciNet  Google Scholar 

  6. T. Kato, Perturbation Theory for Linear Operators [Russian translation], Mir, Moscow (1972).

    MATH  Google Scholar 

  7. A. F. Timan, Theory of Approximation for Functions of Real Variabe [in Russian], Fizmatgiz, Moscow (1960).

    Google Scholar 

  8. H. B. Dwight, Tables of Integrals and Other Mathematical Data, MacMillan, New York (1961).

    Google Scholar 

  9. Yu. A. Brudnyi, S. G. Krein and E. M. Semenov, “Interpolation of linear operators,” in: VINITI Series in Mathematical Analysis [in Russian], Vol. 24, VINITI, Moscow (1986), pp. 3–164.

    Google Scholar 

  10. Ya. V. Radyno, “The space of exponential-type vectors,” Dokl. Akad. Nauk Belorus. SSR, 27, No. 9, 791–793 (1983).

    MATH  MathSciNet  Google Scholar 

  11. N. Dunford and J. T. Schwartz, Linear Operators. Part III. Spectral Operators [Russian translation], Mir, Moscow (1974).

    MATH  Google Scholar 

  12. S. M. Nikol’skii, Approximation of Functions of Many Variables and Imbedding Theorems [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  13. I. I. Ibragimov, Extremal Properties of Entire Functions of Finite DEgree [in Russian], Academy of Sciences of Azerb. SSR, Baku (1962).

    Google Scholar 

  14. G. V. Radzievskii, “Moduli of continuity defined by zero continuation of functions and K-functionals with restrictions,” Ukr. Mat. Zh., 48, No. 11, 1537–1554 (1996).

    Article  MathSciNet  Google Scholar 

  15. S. M. Nikol’skii, “On the Dini-Lipschitz condition of convergence of Fourier series,” Dokl. Akad. Nauk SSSR, 73, No. 3, 457–460 (1950).

    Google Scholar 

  16. E. Hille and R. S. Phillips, Functional Analysis and Semigroups [Russian translation], Inostrannaya Literatur, Moscow (1962).

    Google Scholar 

  17. K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs (1962).

    MATH  Google Scholar 

  18. A. I. Stepanets, Classification and Approximation of Periodic Functions [in Russian], Naukova Dumka, Kiev (1987).

    MATH  Google Scholar 

  19. G. V. Radzievskii, “On linear independence of Keldysh derived chains for operator functions analytic in a half plane,” Mat. Sb., 132, No. 4, 556–577 (1987).

    Google Scholar 

  20. V. K. Dzyadyk, Introduction into the Theory of Uniform Approximation of Functions by Polynomials [in Russian], Nauka, Moscow (1977).

    Google Scholar 

Download references

Authors
  1. G. V. Radzievskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 6, pp. 754–773, June, 1997.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Radzievskii, G.V. On the best approximations and rate of convergence of decompositions in the root vectors of an operator. Ukr Math J 49, 844–864 (1997). https://doi.org/10.1007/BF02513425

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02513425

Keywords

Search

Navigation