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Kolmogorov-Type Inequalities for the Norms of Fractional Derivatives of Functions Defined on the Positive Half Line

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Ukrainian Mathematical Journal Aims and scope

We obtain new Kolmogorov-type sharp inequalities estimating the norm of the Marchaud fractional derivative \( {\left\Vert {D}_{-}^kf\right\Vert}_{\infty } \) of a function f defined on the positive half line in terms of ‖f‖p, 1 < p <  ∞ , and ‖f〞‖1. We also solve the following related problems: the Stechkin problem of the best approximation of the operator \( {D}_{-}^k \) by linear bounded operators and the problem of the best possible recovery of the operator \( {D}_{-}^k \) on a class of elements given with errors.

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References

  1. V. N. Gabushin, “Inequalities for norms of a function and its derivatives in metrics of Lp,” Mat. Zametki, 1, No. 3, 291–298 (1967); English translation: Math. Notes, 1, No 3, 194–198 (1967).

  2. 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 

  3. E. Landau, “Einige Ungleichungen für zweimal differenzierbare Funktion,” Proc. London Math. Soc., 13, 43–49 (1913).

    MATH  Google Scholar 

  4. J. Hadamard, “Sur le module maximum d’une fonction et de ses dérivées,” C. R. Soc. Math. France, 41, 68–72 (1914).

    Google Scholar 

  5. G. H. Hardy and J. E. Littlewood, “Contribution to the arithmetic theory of series,” Proc. London Math. Soc. (2), 11, 411–478 (1912).

    MathSciNet  MATH  Google Scholar 

  6. G. E. Shilov, “On inequalities between derivatives,” in: Collection of Research Works of Students of the Moscow State University [in Russian] (1937), pp. 17–27.

  7. A. N. Kolmogorov, “On inequalities between the upper bounds of the successive derivatives of an arbitrary function on the infinite interval,” Uch. Zap. MGU, Math., 30, No. 1 (1939).

  8. G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge University Press, Cambridge (1934).

    MATH  Google Scholar 

  9. E. M. Stein, “Functions of exponential type,” Ann. Math., 65, No. 3, 582–592 (1957).

    Article  MathSciNet  Google Scholar 

  10. L. V. Taikov, “Kolmogorov-type inequalities and the best formulas of numerical differentiation,” Mat. Zametki, 4, No. 2, 233–238 (1968).

    MathSciNet  Google Scholar 

  11. A. P. Matorin, “On inequalities between the maximum values of moduli of a function and its derivatives on a half line,” Ukr. Mat. Zh., 7, No. 2, 262–266 (1955).

    MathSciNet  Google Scholar 

  12. I. J. Shoenberg and A. Cavaretta, Solution of Landau’s Problem, Concerning Higher Derivatives on Half-Line, MRC Technical Summary Report 1050, Madison, Wisconsin (1970).

    Google Scholar 

  13. Yu. I. Lyubich, “On inequalities between the degrees of linear operators,” Izv. Akad. Nauk SSSR, Ser. Mat., 24, 825–864 (1960).

  14. N. P. Kuptsov, “Kolmogorov estimates for the derivatives in L2[0,∞),” Tr. Mat. Inst. Akad. Nauk SSSR, 138, 94–117 (1975).

  15. V. N. Gabushin, “On the best approximation of differential operators on the semiaxis,” Mat. Zametki, 6, No. 3, 573–582 (1969).

    MathSciNet  Google Scholar 

  16. N. P. Kuptsov, “On the exact constants in inequalities between the norms of functions and their derivatives,” Mat. Zametki, 41, No. 3, 313–319 (1987).

    MathSciNet  Google Scholar 

  17. G. A. Kalyabin, “Exact constants in inequalities for intermediate derivatives (Gabushin’s case),” Funkts. Anal. Prilozhen., 38, No. 3, 29–38 (2004).

    Article  Google Scholar 

  18. V. V. Arestov and V. N. Gabushin, “Best approximation of unbounded operators by bounded ones,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., 42–68 (1995); English translation: Russian Math. (Izv. VUZ), 38–63 (1995).

  19. K. B. Oldham and J. Spanier, ”The fractional calculus; theory and applications of differentiation and integration to arbitrary order,” in: Mathematics in Science and Engineering, Vol. 111, Academic Press, New York; London (1974).

  20. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Taylor & Francis, London (2002).

    MATH  Google Scholar 

  21. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London (2010).

    Book  Google Scholar 

  22. V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Berlin (2010).

    Book  Google Scholar 

  23. A. Marchaud, “Sur les dérivées et sur les différences des fonctions de variables réelles,” J. Math. Pures Appl. (9), 6, 337–426 (1927).

  24. S. P. Geisberg, “Generalization of Hadamard’s inequality,” Sb. Nauch. Trud. Leningrad. Mekh. Inst., 50, 42–54 (1965).

    MathSciNet  Google Scholar 

  25. V. V. Arestov, “Inequalities for fractional derivatives on the half-line,” in: Approximation Theory, Vol. 4, Banach Center Publ.,Warsaw (1979), pp. 19–34.

  26. A. P. Buslaev and V. M. Tikhomirov, “On inequalities for the derivatives in the multidimensional case,” Mat. Zametki, 25, No. 1, 59–73 (1979).

    MathSciNet  MATH  Google Scholar 

  27. G. G. Magaril-Il’yaev and V. M. Tihomirov, “On the Kolmogorov inequality for fractional derivatives on the half-line,” Anal. Math., 7, No. 1, 37–47 (1981).

    Article  MathSciNet  Google Scholar 

  28. V. F. Babenko and M. S. Churilova, “On Kolmogorov-type inequalities for fractional derivatives,” Visn. Dnipropetr. Univ., Ser. Mat., 6, 16–20 (2001).

  29. M. S. Churilova, Kolmogorov-Type Inequalities for Fractional Derivatives and Their Applications in Approximation Theory [in Ukrainian], Candidate-Degree Thesis (Physics and Mathematics), Dnipropetrovsk (2006).

  30. V. F. Babenko and M. S. Churilova, “On the Kolmogorov-type inequalities for fractional derivatives defined on the real axis,” Visn. Dnipropetr. Univ., Ser. Mat., 13, 28–34 (2008).

  31. V. F. Babenko and N. F. Parfinovich, “Kolmogorov-type inequalities for the norms of Riesz derivatives of functions of many variables and some applications,” Tr. Inst. Mat. Mekh. Ural. Otdel. Ros. Akad. Nauk, 17, No. 3, 60–70 (2011).

    Google Scholar 

  32. V. F. Babenko, N. F. Parfinovich, and S. A. Pichugov, “Kolmogorov-type inequalities for the norms of Riesz derivatives of functions of many variables with Laplacian bounded in L∞ and some related problems,” Mat. Zametki, 95, No. 1, 3–17 (2014).

    Article  Google Scholar 

  33. V. F. Babenko, M. S. Churilova, N. V. Parfinovych, and D. S. Skorokhodov, “Kolmogorov type inequalities for the Marchaud fractional derivatives on the real line and the half-line,” J. Inequal. Appl., Article No. 504 (2014).

  34. V. P. Motornyi, V. F. Babenko, A. A. Dovgoshei, and O. I. Kuznetsova, Approximation Theory and Harmonic Analysis [in Russian], Naukova Dumka, Kiev (2012).

    Google Scholar 

  35. S. B. Stechkin, “Best approximation of linear operators,” Mat. Zametki, 1, No. 2, 137–148 (1967); English translation: Math. Notes, 1, No. 2, 91–99 (1967).

  36. S. M. Nikol’skii, A Course in Mathematical Analysis [in Russian], Nauka, Moscow (2001).

    Google Scholar 

  37. M. A. Krasnosel’skii, A. I. Perov, A. I. Povolotskii, and P. P. Zabreiko, Vector Fields in the Plane [in Russian], Nauka, Moscow (1963).

    Google Scholar 

  38. V. N. Gabushin, “Best approximations of functionals on some sets,” Mat. Zametki, 8, No. 5, 551–562 (1970).

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. O. Honchar Dnipro National University, Dnipro, Ukraine

    O. Kozynenko & D. Skorokhodov

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  1. O. Kozynenko
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  2. D. Skorokhodov
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Correspondence to O. Kozynenko.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 10, pp. 1372–1385, October, 2020. Ukrainian DOI: 10.37863/umzh.v72i10.1074.

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Kozynenko, O., Skorokhodov, D. Kolmogorov-Type Inequalities for the Norms of Fractional Derivatives of Functions Defined on the Positive Half Line. Ukr Math J 72, 1579–1594 (2021). https://doi.org/10.1007/s11253-021-01873-7

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  • DOI: https://doi.org/10.1007/s11253-021-01873-7

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