Skip to main content
SpringerLink
Log in

Jackson-Type Inequalities for the Special Moduli of Continuity on the Entire Real Axis and the Exact Values of Mean ν - Widths for the Classes of Functions in the Space L 2 (ℝ)

  • Published:
Ukrainian Mathematical Journal Aims and scope

The exact values of constants are obtained in the space L 2(ℝ) for the Jackson-type inequalities for special moduli of continuity of the k th order defined by the Steklov operator S h (\( f \)) instead of the translation operator T h (\( f \)) in the case of approximation by entire functions of exponential type σ ∈ (0,∞) . The exact values of the mean ν -widths (linear, Bernstein, and Kolmogorov) are also obtained for the classes of functions defined by the indicated characteristic of smoothness.

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. S. N. Bernstein, “On the best approximation of continuous functions on the entire real axis with the use of entire functions of given degree” (1912); in: Collected Works, Vol. 2 [in Russian], Izd. Akad. Nauk SSSR, Moscow (1952), pp. 371–375.

  2. N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Gostekhizdat, Moscow (1947).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

  9. A. I. Stepanets, “Classes of functions defined on the real line and their approximations by entire functions. I,” Ukr. Mat. Zh., 42, No. 1,102–112 (1990); English translation: Ukr. Math. J., 42, No. 1, 93–102 (1990).

  10. A. I. Stepanets, “Classes of functions defined on the real axis and their approximations by entire functions. II,” Ukr. Mat. Zh., 42, No. 2, 210–222 (1990); English translation: Ukr. Math. J., 42, No. 2, 186–197 (1990).

  11. 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 

  12. A. A. Ligun and V. G. Doronin, “Exact constants in Jackson-type inequalities for L 2-approximation on an axis,” Ukr. Mat. Zh., 61, No. 1, 92–98 (2009); English translation: Ukr. Math. J., 61, No. 1, 112–120 (2009).

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

    Google Scholar 

  14. S. B. Vakarchuk and V. G. Doronin, “Best mean-square approximations by entire functions of finite degree on a straight line and exact values of mean widths of functional classes,” Ukr. Mat. Zh., 62, No. 8, 1032–1043 (2010); English translation: Ukr. Math. J., 62, No. 8, 1199–1212 (2011).

  15. S. Ya. Yanchenko, “Approximation of the classes S p r , θ B(ℝd) of functions of many variables by entire functions of a special form,” Ukr. Mat. Zh., 62, No. 8, 1124–1138 (2010); English translation: Ukr. Math. J., 62, No. 8, 1307–1325 (2011).

  16. S. B. Vakarchuk, “On some extremal problems of the approximation theory of functions on the real axis. I,” Ukr. Mat. Visn., 9, No. 3, 401–429 (2012).

    Google Scholar 

  17. S. B. Vakarchuk, “On some extremal problems of the approximation theory of functions on the real axis. II,” Ukr. Mat. Visn., 9, No. 4, 578–602 (2012).

    MathSciNet  Google Scholar 

  18. S. B. Vakarchuk and V. I. Zabutnaya, “Exact inequality of the Jackson–Stechkin type in L 2 and the widths of functional classes,” Mat. Zametki, 86, No. 3, 328–336 (2009).

    Article  MathSciNet  Google Scholar 

  19. S. B. Vakarchuk and V. I. Zabutnaya, “Inequalities of the Jackson–Stechkin type for special moduli of continuity and the widths of functional classes in the space L 2 ,” Mat. Zametki, 92, No. 4, 497–514 (2012).

    Article  MathSciNet  Google Scholar 

  20. S. B. Vakarchuk and V. I. Zabutnaya, “On the best polynomial approximation in the space L 2 and the widths of some classes of functions,” Ukr. Mat. Zh., 64, No. 8, 1025–1032 (2012); English translation: Ukr. Math. J., 64, No. 8, 1168–1176 (2013).

  21. V. A. Abilov and F. V. Abilova, “Some problems of approximation of 2\( \pi \)-periodic functions by Fourier sums in the space L 2(2\( \pi \)),” Mat. Zametki, 76, No. 6, 803–811 (2004).

    Article  MathSciNet  Google Scholar 

  22. V. Kokilashvili and Y. E. Yildirir, “On the approximation in weighted Lebesgue space,” in: Proc. of the Razmadze Mathematical Institute, 143 (2007), pp. 103–113.

    MATH  MathSciNet  Google Scholar 

  23. R. Akg¨un, “Sharp Jackson and converse theorem of trigonometric approximation in weighted Lebesgue spaces,” in: Proc. of the Razmadze Mathematical Institute, 152 (2010), pp. 1–18.

  24. Ya. I. Khurgin and V. P. Yakovlev, Finite Functions in Physics and Engineering [in Russian], Nauka, Moscow (1971).

    Google Scholar 

  25. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford (1948).

    Google Scholar 

  26. E. F. Beckenbach and R. Bellman, Inequalities, Springer, Berlin (1961).

    Book  Google Scholar 

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

    Google Scholar 

  28. N. I. Chernykh, “On the best approximation of periodic functions by trigonometric polynomials in L 2 ,” Mat. Zametki, 2, No. 5, 513–522 (1967).

    MathSciNet  Google Scholar 

  29. S. B. Vakarchuk and V. I. Zabutnaya, “Some problems of the approximation theory of 2\( \pi \)-periodic functions in the spaces L p , 1 ≤ p ≤ ∞,” in: Problems of the Approximation Theory of Functions and Related Problems, Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, 1, No. 1 (2004), pp. 25–41.

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

  32. V. M. Tikhomirov, “On the approximate characteristics of smooth functions of many variables,” in: Theory of Cubature Formulas and Numerical Mathematics [in Russian], Nauka, Novosibirsk (1980), pp. 183–188.

Download references

Author information

Authors and Affiliations

  1. Nobel Dnepropetrovsk University, Dnepropetrovsk, Ukraine

    S. B. Vakarchuk

Authors
  1. S. B. Vakarchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 6, pp. 740–766, June, 2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vakarchuk, S.B. Jackson-Type Inequalities for the Special Moduli of Continuity on the Entire Real Axis and the Exact Values of Mean ν - Widths for the Classes of Functions in the Space L 2 (ℝ). Ukr Math J 66, 827–856 (2014). https://doi.org/10.1007/s11253-014-0977-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-014-0977-9

Keywords

Search

Navigation