Skip to main content
SpringerLink
Log in

Best mean square approximations by entire functions of finite degree on a straight line and exact values of mean widths of functional classes

  • Published:
Ukrainian Mathematical Journal Aims and scope

We obtain exact Jackson-type inequalities in the case of the best mean square approximation by entire functions of finite degree ≤ σ on a straight line. For classes of functions defined via majorants of averaged smoothness characteristics Ω1(f, t ), t > 0, we determine the exact values of the Kolmogorov mean ν-width, linear mean ν-width, and Bernstein mean ν-width, ν > 0.

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

    Google Scholar 

  2. N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  3. I. I. Ibragimov, Theory of Approximation by Entire Functions [in Russian], Élm, Baku (1979).

    MATH  Google Scholar 

  4. I. I. Ibragimov and F. G. Nasibov, “On the estimation of the best approximation of a summable function on the real axis by entire functions of finite degree,” Dokl. Akad. Nauk SSSR, 194, No. 5, 1013–1016 (1970).

    MathSciNet  Google Scholar 

  5. F. G. Nasibov, “On approximation by entire functions in L 2,” Dokl. Akad. Nauk Azerb. SSR, 42, No. 4, 3–6 (1986).

    MATH  MathSciNet  Google Scholar 

  6. V. Yu. Popov, “On the best mean square approximation by entire functions of exponential type,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 6, 65–73 (1972).

  7. V. K. Dzyadyk, “On the least upper bounds of the best approximations on some classes of continuous functions defined on the real axis,” Dopov. Akad. Nauk Ukr. SSR, Ser. A, No. 7, 589–592 (1975).

    Google Scholar 

  8. A. Yu. Gromov, “On exact constants of approximation of differentiable functions by entire functions,” in: Investigations in Contemporary Problems of Summation and Approximation of Functions and Their Applications [in Russian], Issue 7 (1976), pp. 17–21.

  9. S. M. Nikol’skii, Approximation of Functions of Many Variables and Imbedding Theorems [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  10. S. B. Vakarchuk, “Exact constant in an inequality of Jackson type for L 2-approximation on the line and exact values of mean widths of functional classes,” East J. Approxim., 10, No. 1–2, 27–39 (2004).

    MATH  MathSciNet  Google Scholar 

  11. V. G. Doronin and A. A. Ligun, “On exact inequalities of Jackson type for entire functions in L 2,” Visn. Dnipropetr. Univ., Ser. Mat., No. 8, 89–93 (2007).

    Google Scholar 

  12. S. B. Vakarchuk and M. B. Vakarchuk, “On the best mean square approximation by entire functions of finite degree on a straight line,” Visn. Dnipropetr. Univ., Ser. Mat., 17, No. 6/1, 36–41 (2009).

    Google Scholar 

  13. K. V. Runovskii, “On approximation by families of linear polynomial operators in the space L p , 0 < p < 1,” Mat. Sb., 185, No. 8, 81–102 (1994).

    Google Scholar 

  14. É. A. Storozhenko, V. G. Krotov, and P. Oswald, “Direct and inverse theorems of Jackson type in the space L p , 0 < p < 1,” Mat. Sb., 98, No. 3, 395–415 (1975).

    MathSciNet  Google Scholar 

  15. S. B. Vakarchuk, “Exact constants in inequalities of Jackson type and exact values of widths of functional classes from L 2,” Mat. Zametki, 78, No. 5, 792–796 (2005).

    MathSciNet  Google Scholar 

  16. S. B. Vakarchuk and V. I. Zabutna, “Widths of functional classes from L 2 and exact constants in Jackson-type inequalities,” East J. Approxim., 14, No. 4, 411–421 (2008).

    MathSciNet  Google Scholar 

  17. A. A. Ligun, “Some inequalities between the best approximations and moduli of continuity in the space L 2,” Mat. Zametki, 24, No. 6, 785–792 (1978).

    MATH  MathSciNet  Google Scholar 

  18. V. D. Rybasenko and I. D. Rybasenko, Elementary Functions. Formulas, Tables, and Graphs [in Russian], Nauka, Moscow (1987).

    Google Scholar 

  19. G. G. Magaril-Il’yaev, “φ-Mean widths of classes of functions on a straight line,” Usp. Mat. Nauk, 45, No. 2, 211–212 (1990).

    MathSciNet  Google Scholar 

  20. V. M. Tikhomirov, “On approximation characteristics of smooth functions of many variables,” in: Theory of Cubature Formulas and Computational Mathematics, Proceedings of the Conference on Differential Equations and Computational Mathematics (Novosibirsk, 1978) [in Russian], Nauka, Novosibirsk (1980), pp. 183–188.

  21. G. G. Magaril-Il’yaev, “Mean dimensionality, widths, and optimal reconstruction of Sobolev classes of functions on a straight line,” Mat. Sb., 182, No. 11, 1635–1656 (1991).

    Google Scholar 

  22. G. G. Magaril-Il’yaev, “Mean dimensionality and widths of functions on a straight line,” Dokl. Akad. Nauk SSSR, 318, No. 1, 35–38 (1991).

    MathSciNet  Google Scholar 

  23. I. M. Glazman and Yu. I. Lyubich, Finite-Dimensional Linear Analysis [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  24. S. B. Vakarchuk, “On strong asymptotics of mean N-widths of classes of functions analytic on a real straight line,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 1, 1–4 (1996).

Download references

Author information

Authors and Affiliations

  1. Dnepropetrovsk University of Economics and Law, Dnepropetrovsk, Ukraine

    S. B. Vakarchuk

  2. Gonchar Dnepropetrovsk National University, Dnepropetrovsk, Ukraine

    V. G. Doronin

Authors
  1. S. B. Vakarchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. G. Doronin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 8, pp. 1032–1043, August, 2010.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vakarchuk, S.B., Doronin, V.G. Best mean square approximations by entire functions of finite degree on a straight line and exact values of mean widths of functional classes. Ukr Math J 62, 1199–1212 (2011). https://doi.org/10.1007/s11253-011-0424-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-011-0424-0

Keywords

Search

Navigation