Advertisement

Ukrainian Mathematical Journal

, Volume 70, Issue 9, pp 1345–1374 | Cite as

Generalized Characteristics of Smoothness and Some Extreme Problems of the Approximation Theory of Functions in the Space L2(ℝ). I

  • S. B. Vakarchuk
Article
  • 21 Downloads

We consider the generalized characteristics of smoothness of the functions ωw(f, t) and Λw(f, t), t > 0, in the space L2(ℝ) and on the classes \( {L}_2^{\alpha } \) (ℝ) defined with the help of fractional-order derivatives α ϵ (0, ∞) and obtain the exact Jackson-type inequalities for ωw(f).

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 via entire functions of a given power (1912),” in: Collection of Works [in Russian], Vol. 2, Akad. Nauk SSSR, Moscow (1952), pp. 371–375.Google Scholar
  2. 2.
    N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Gostekhizdat, Moscow (1947).zbMATHGoogle Scholar
  3. 3.
    A. F. Timan, Approximation Theory of Functions of Real Variable [in Russian], Fizmatgiz, Moscow (1960).Google Scholar
  4. 4.
    M. F. Timan, “Approximation of functions given on the entire real axis by entire functions of exponential type,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 2, 89–101 (1968).Google Scholar
  5. 5.
    S. M. Nikol’skii, Approximation of Functions of Many Variables and Embedding Theorems [in Russian], Nauka, Moscow (1977).Google Scholar
  6. 6.
    I. I. Ibragimov and F. G. Nasibov, “On the estimation of the best approximation of a summable function on the real axis via entire functions of finite power,” 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).zbMATHGoogle 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.
    V. G. Ponomarenko, “Fourier integrals and the best approximation by entire functions,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 3, 109–123 (1966).Google Scholar
  10. 10.
    G. Gaimnazarov, “On the moduli of continuity of fractional order for functions given on the entire real axis,” Dokl. Akad. Nauk Tadzhik. SSR, 24, No. 3, 148–150 (1981).MathSciNetGoogle Scholar
  11. 11.
    G. Gaimnazarov, “Some relations for the moduli of continuity of fractional order in the space L p(, ∞),Izv. Akad. Nauk Tadzhik. SSR, No. 3, 8–13 (1985).Google Scholar
  12. 12.
    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
  13. 13.
    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
  14. 14.
    A. A. Ligun and V. G. Doronin, “Exact constants in Jackson-type inequalities for L2-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
  15. 15.
    V. V. Arestov, “On Jackson inequalities for approximation in L 2 of periodic functions by trigonometric polynomials and of functions on the line by entire functions,” in: Approximation Theory: A Volume Dedicated to Borislaw Bojanov, Marin Drinov Acad. Publ. House, Sofia (2004), pp. 1–19.Google Scholar
  16. 16.
    A. G. Babenko, “Exact Jackson–Stechkin inequality in the space L 2(ℝm),” in: Proc. of the Institute of Mathematics and Mechanics, Ural Division of the Russian Academy of Sciences [in Russian], No. 5 (1998), pp. 3–7.Google Scholar
  17. 17.
    C. N. Vasil’ev, “Jackson inequality in L 2(ℝN) with generalized modulus of continuity,” in: Proc. of the Institute of Mathematics and Mechanics, Ural Division of the Russian Academy of Sciences [in Russian], 16, No. 4 (2010), pp. 93–99.Google Scholar
  18. 18.
    S. B. Vakarchuk, “Exact constant in an inequality of Jackson type for L2-approximation on the line and exact values of mean widths of functional classes,” East J. Approxim., 10, No. 1-2, 27–39 (2004).zbMATHGoogle Scholar
  19. 19.
    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
  20. 20.
    S. B. Vakarchuk, “Best mean-square approximation of functions defined on the real axis by entire functions of exponential type,” Ukr. Mat. Zh., 64, No. 5, 604–615 (2012); English translation: Ukr. Math. J., 64, No. 5, 680–692 (2012).Google Scholar
  21. 21.
    S. B. Vakarchuk, “On some extreme problems of approximation theory of functions on the real axis. I,” J. Math. Sci., 188, No. 2, 146–166 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    S. B. Vakarchuk, “On some extremal problems of approximation theory of functions on the real axis. II,” J. Math. Sci., 190, No. 4, 613–630 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    M. Sh. Shabozov and G. A. Yusupov, “On the exact values of mean -widths for some classes of entire functions,” in: Proc. of the Institute of Mathematics and Mechanics, Ural Division of the Russian Academy of Sciences [in Russian], 18, No. 4 (2012), pp. 315–327.Google Scholar
  24. 24.
    G. A. Yusupov, “On the best mean-square approximations on the entire axis by entire functions of exponential type,” Dokl. Akad. Nauk Resp. Tadzhikistan, 56, No. 3, 192–195 (2013).MathSciNetGoogle Scholar
  25. 25.
    S. B. Vakarchuk, “Jackson-type inequalities for the special moduli of continuity on the entire real axis and the exact values of mean -widths for the classes of functions in the space L 2(ℝ),Ukr. Mat. Zh., 66, No. 6, 740–766 (2014); English translation: Ukr. Math. J., 66, No. 6, 827–856 (2014).Google Scholar
  26. 26.
    S. B. Vakarchuk, “Best mean-square approximations by entire functions of exponential type and mean -widths of classes of functions on the line,” Math. Notes, 96, No. 6, 878–896 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    S. B. Vakarchuk, M. Sh. Shabozov, and M. R. Langarshoev, “On the best mean square approximations by entire functions of exponential type in L 2(ℝ) and mean ν-widths of some functional classes,” Russian Math., 58, No. 7, 25–41 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    S. Ya. Yanchenko, “Approximations of classes \( {B}_{p,\theta}^{\varOmega } \) of functions of many variables by entire functions in the space L q(ℝd),Ukr. Mat. Zh., 62, No. 1, 123–135 (2010); English translation: Ukr. Math. J., 62, No. 1, 136–150 (2010).Google Scholar
  29. 29.
    S. B. Vakarchuk, “Mean-square approximation of function classes, given on the all real axis ℝ by the entire functions of exponential type,” Int. J. Adv. Math., 6, 1–12 (2016).CrossRefGoogle Scholar
  30. 30.
    S. B. Vakarchuk, “Exact constants in Jackson-type inequalities for the best mean square approximation in L 2(ℝ) and exact values of mean -widths of the classes of functions,” J. Math. Sci., 224, No. 4, 582–603 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    S. Yu. Artamonov, “Nonperiodic modulus of smoothness corresponding to the Riesz derivative,” Math. Notes, 99, No. 6, 928–931 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    S. B. Vakarchuk, “On the moduli of continuity and fractional-order derivatives in the problems of best mean-square approximations by entire functions of the exponential type on the entire real axis,” Ukr. Mat. Zh., 69, No. 5, 599–623 (2017); English translation: Ukr. Math. J., 69, No. 5, 696–724 (2017).Google Scholar
  33. 33.
    S. B. Vakarchuk, “Jackson-type inequalities with generalized modulus of continuity and exact values of the n-widths for the classes of (ψ, β)-differentiable functions in L 2 . I,” Ukr. Mat. Zh., 68, No 6, 723–745 (2016); English translation: Ukr. Math. J., 68, No 6, 823–848 (2016).Google Scholar
  34. 34.
    S. B. Vakarchuk, “Jackson-type inequalities with generalized modulus of continuity and exact values of the n-widths for the classes of ( , β)-differentiable functions in L2. II,” Ukr. Mat. Zh., 68, No. 8, 1021–1036 (2016); English translation: Ukr. Math. J., 68, No. 8, 1165–1183 (2017).Google Scholar
  35. 35.
    S. B. Vakarchuk, “Jackson-type inequalities with generalized modulus of continuity and exact values of the n-widths for the classes of ( , β)-differentiable functions in L2. III,” Ukr. Mat. Zh., 68, No 10, 1299–1319 (2016); English translation: Ukr. Math. J., 68, No 10, 1495–1518 (2017).Google Scholar
  36. 36.
    Z. Ditzian and V. Totik, Moduli of Smoothness, Springer, New York (1987).CrossRefzbMATHGoogle Scholar
  37. 37.
    K. Runovski and H.-J. Schmeisser, On Modulus of Continuity Related to Riesz Derivative, Preprint, Friedrich Schiller University, Jena (2011).zbMATHGoogle Scholar
  38. 38.
    J. Boman and H. S. Shapiro, “Comparison theorems for a generalized modulus of continuity,” Ark. Mat., 9, No. 1, 91–116 (1971).MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    J. Boman, “Equivalence of generalized moduli of continuity,” Ark. Mat., 18, No. 1, 73–100 (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    C. N. Vasil’ev, “Exact Jackson–Stechkin inequality in L 2 with modulus of continuity generated by an arbitrary finite-difference operator with constant coefficients,” Dokl. Akad. Nauk, 385, No. 1, 11–14 (2002).MathSciNetGoogle Scholar
  41. 41.
    C. N. Vasil’ev, “Widths of some classes of functions in the space L 2 on a period,” in: Proc. of the Institute of Mathematics and Mechanics, Ural Division of the Russian Academy of Sciences [in Russian], 19, No. 4 (2013), pp. 42–47.Google Scholar
  42. 42.
    A. I. Kozko and A. V. Rozhdestvenskii, “On Jackson’s inequality for generalized moduli of continuity,” Math. Notes, 73, No. 5, 736–741 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    D. V. Gorbachev, “Estimation of the optimal argument in the exact multidimensional Jackson–Stechkin L2-inequality,” in: Proc. of the Institute of Mathematics and Mechanics, Ural Division of the Russian Academy of Sciences [in Russian], 20, No. 1, (2014), pp. 83–91.Google Scholar
  44. 44.
    A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series [in Russian], Vol. 1, Nauka, Moscow (1981).zbMATHGoogle Scholar
  45. 45.
    P. L. Butzer, H. Dyckhoff, E. Gorlich, and R. L. Stens, “Best trigonometric approximation, fractional order derivatives and Lipschitz classes,” Canad. J. Math., l29, No. 4, 781–793 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Some Their Applications [in Russian], Nauka i Tekhnika, Minsk (1987).zbMATHGoogle Scholar
  47. 47.
    P. L. Butzer and U. Westphal, “An introduction to fractional calculus,” in: Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000), pp. 1–85.CrossRefzbMATHGoogle Scholar
  48. 48.
    Ya. I. Khurgin and V. P. Yakovlev, Finite Functions in Physics and Engineering [in Russian], Nauka, Moscow (1971).zbMATHGoogle Scholar
  49. 49.
    N. I. Akhiezer, Lectures on Integral Transformations [in Russian], Vyshcha Shkola, Kharkov (1984).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • S. B. Vakarchuk
    • 1
  1. 1.Nobel Dnepr UniversityDneprUkraine

Personalised recommendations