Ukrainian Mathematical Journal

, Volume 71, Issue 2, pp 179–189 | Cite as

One Inequality of the Landau–Kolmogorov Type for Periodic Functions of Two Variables

  • V. F. BabenkoEmail author

We obtain a new sharp inequality of the Landau–Kolmogorov type for a periodic function of two variables estimating the convolution of the best uniform approximations of its partial primitives by the sums of functions of single variable via the L-norm of the function itself and uniform norms of its mixed primitives. Some applications of the obtained inequality are also presented.


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  1. 1.
    V. V. Arestov and V. N. Gabushin, “Best approximation of unbounded operators by bounded operators,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., 11, 44–66 (1995).zbMATHGoogle Scholar
  2. 2.
    V. V. Arestov, “Approximation of unbounded operators by bounded operators and related extremal problems,” Usp. Mat. Nauk, No. 6, 88–124 (1996).Google Scholar
  3. 3.
    V. F. Babenko, N. P. Korneichuk, V. A. Kofanov, and S. A. Pichugov, Inequalities for Derivatives and Their Applications [in Russian], Naukova Dumka, Kiev (2003).Google Scholar
  4. 4.
    V. N. Konovalov, “Sharp inequalities for the norms of functions, third partial and second mixed derivatives,” Mat. Zametki, 23, No. 1, 67–78 (1978).MathSciNetGoogle Scholar
  5. 5.
    O. A. Timoshchin, “Sharp inequalities between the norms of the second and third derivatives,” Dokl. Ros. Akad. Nauk, 344, No. 1, 20–22 (1995).Google Scholar
  6. 6.
    V. F. Babenko, “On sharp Kolmogorov-type inequalities for functions of two variables,” Dop. Nats. Akad. Nauk Ukr., No 5, 7–11 (2000).Google Scholar
  7. 7.
    V. V. Arestov, “Some extremal problems for differentiable functions of one variable,” Tr. Mat. Inst. Akad. Nauk SSSR, 138, 3–26 (1975).MathSciNetGoogle Scholar
  8. 8.
    B. E. Klots, “Approximation of differentiable functions by functions of higher smoothness,” Mat. Zametki, 21, No. 1, 21–32 (1977).MathSciNetzbMATHGoogle Scholar
  9. 9.
    A. A. Ligun, “Inequalities for upper bounds of functionals,” Anal. Math., 2, No. 1, 11–40 (1976).MathSciNetCrossRefGoogle Scholar
  10. 10.
    N. P. Korneichuk, A. A. Ligun, and V. G. Doronin, Approximation with Restrictions [in Russian], Naukova Dumka, Kiev (1982).Google Scholar
  11. 11.
    V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Multivariate inequalities of Kolmogorov type and their applications,” in: Proc. of the Conference “Multivariate Approximation and Splines,” (Mannheim, September 7–10, 1996) (1997), pp. 1–12.Google Scholar
  12. 12.
    N. P. Korneichuk, Extremal Problems of Approximation Theory [in Russian], Nauka, Moscow (1976).Google Scholar
  13. 13.
    V. M. Tikhomirov, “Convex analysis,” in: VINITI Series in Contemporary Problems of Mathematics (Fundamental Trends) [in Russian], Vol. 14, VINITI, Moscow (1987), pp. 5–101.Google Scholar

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Authors and Affiliations

  1. 1.Dnepr National UniversityDneprUkraine

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