Advertisement

Ukrainian Mathematical Journal

, Volume 66, Issue 6, pp 827–856 | Cite as

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)

  • S. B. Vakarchuk
Article

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.

Keywords

Entire Function Approximation Theory Exponential Type Function Sinc Require Equality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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.Google Scholar
  2. 2.
    N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Gostekhizdat, Moscow (1947).Google Scholar
  3. 3.
    A. F. Timan, Approximation Theory of Functions of Real Variable [in Russian], Fizmatgiz, Moscow (1960).Google Scholar
  4. 4.
    S. M. Nikol’skii, Approximation of Functions of Many Variables and Imbedding Theorems [in Russian], Nauka, Moscow (1969).Google Scholar
  5. 5.
    I. I. Ibragimov, Approximation Theory by Entire Functions [in Russian], Élm, Baku (1979).Google Scholar
  6. 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).MathSciNetGoogle Scholar
  7. 7.
    F. G. Nasibov, “On the approximation by entire functions in L 2 ,Dokl. Akad. Nauk Azerb. SSR, 42, No. 4, 3–6 (1986).zbMATHMathSciNetGoogle Scholar
  8. 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).Google Scholar
  9. 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).Google Scholar
  10. 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).Google Scholar
  11. 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).zbMATHMathSciNetGoogle Scholar
  12. 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).Google Scholar
  13. 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. 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).Google Scholar
  15. 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).Google Scholar
  16. 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. 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).MathSciNetGoogle Scholar
  18. 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).CrossRefMathSciNetGoogle Scholar
  19. 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).CrossRefMathSciNetGoogle Scholar
  20. 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).Google Scholar
  21. 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).CrossRefMathSciNetGoogle Scholar
  22. 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.zbMATHMathSciNetGoogle Scholar
  23. 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.Google Scholar
  24. 24.
    Ya. I. Khurgin and V. P. Yakovlev, Finite Functions in Physics and Engineering [in Russian], Nauka, Moscow (1971).Google Scholar
  25. 25.
    E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford (1948).Google Scholar
  26. 26.
    E. F. Beckenbach and R. Bellman, Inequalities, Springer, Berlin (1961).CrossRefGoogle Scholar
  27. 27.
    V. D. Rybasenko and I. D. Rybasenko, Elementary Functions. Formulas, Tables, and Graphs [in Russian], Nauka, Moscow (1987).Google Scholar
  28. 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).MathSciNetGoogle Scholar
  29. 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.Google Scholar
  30. 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. 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).MathSciNetGoogle Scholar
  32. 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.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • S. B. Vakarchuk
    • 1
  1. 1.Nobel Dnepropetrovsk UniversityDnepropetrovskUkraine

Personalised recommendations