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Investigations of dnepropetrovsk mathematicians related to inequalities for derivatives of periodic functions and their applications

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We present a survey of investigations of Dnepropetrovsk mathematicians related to Kolmogorov-type exact inequalities for norms of intermediate derivatives of periodic functions and their applications in approximation theory.

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References

  1. G. H. Hardy and J. E. Littlewood, “Contribution to the arithmetic theory of series,” Proc. London Math., 11, No. 2, 411–478 (1912).

    Google Scholar 

  2. E. Landau, “Einige Ungleichungen fur zweimal differenzierbare Functionen,” Proc. London Math.. 13, 43–9 (1913).

    Article  Google Scholar 

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

    Google Scholar 

  4. A. N. Kolmogorov, Selected Works. Mathematics. Mechanics [in Russian] Nauka, Moscow 1985.

    Google Scholar 

  5. Yu. G. Bosse G. E. Shilov, “On inequalities for derivatives,” in: Collection of Works of Student Scientific Societies of MoscowUniversity [in Russian], Moscow University, Moscow (1937), pp. 17–27.

    Google Scholar 

  6. N. P. Komeichuk, V. F. Babenko, and A. A. Ligun, Extremal Properties of Polynomials and Splines [in Russian], Naukova Dumka,Kiev (1992).

    Google Scholar 

  7. N. P. Komeichuk, A. A. Ligun, and V. F. Babenko, Extremal Properties of Polynomials and Splines. Nova, New York (1996).

    Google Scholar 

  8. N. P. Komeichuk, Exact Constants in Approximation Theory [in Russian] Nauka, Moscow 1987.

    Google Scholar 

  9. N. P. Komeichuk, A. A. Ligun, and V. G. Doronin, Approximation with Constraints [in Russian] Naukova Dumka, Kiev 1982.

    Google Scholar 

  10. S. Mandelbrojt, Series Adherentes, Regularisations des Suites. Applications. Gauthier-Villars (1952).

  11. A. A. Ligun, “Exact inequalities for upper bounds of seminorms on the classes of periodic functions,” Mat. Zametki, 13, No. 5, 647–654 (1973).

    MATH  MathSciNet  Google Scholar 

  12. A. A. Ligun, “Exact constants in Jackson-type inequalities,” in: Special Problems in the Theory of Approximation and Optimal Control by Distributed Systems [in Russian], Vyshcha Shkola, Kiev (1990), pp. 5–74.

    Google Scholar 

  13. G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge (1934).

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

    Article  Google Scholar 

  15. A. G. Taikov, “Kolmogorov-type inequalities and the best formulas for numerical differentiation,” Mat. Zametki, 4, No. 5, 233–238 (1968).

    MathSciNet  Google Scholar 

  16. A. P. Matorin, “Inequalities for the largest absolute values of a function and its derivatives on a half line,” Ukr. Mat. Zh., 5, No. 3, 262–266 (1955).

    MathSciNet  Google Scholar 

  17. I. J. Shoenberg and A. Cavaretta, “Solution of Landau’s problem concerning higher derivatives on a half line,” in: Proceedings ofthe Conference on Approximation Theory (Varna-Sofia), Sofia (1972), pp. 297–308.

  18. Yu. I. Lyubich, “On inequalities for powers of a linear operator,” Izv. Akad. Nauk SSSR, Ser. Mat., 24, 825–864 (1960).

    MATH  MathSciNet  Google Scholar 

  19. N.P. Kuptsov, “Kolmogorov estimates for derivatives in L 2[0, ∞),” Tr. Mat. Inst. Akad. Nauk SSSR, 138, 94–117 (1975).

    MATH  Google Scholar 

  20. V. N. Gabushin, “On the best approximation of the operator of differentiation on a half line,” Mat. Zametki, 6, No. 5, 573–582 (1969).

    MathSciNet  Google Scholar 

  21. V. V. Arestov and V. N. Gabushin, “The best approximation of unbounded operators by bounded operators,” Izv. Vyssh. Uchebn.Zaved., Ser. Mat., No. 11, 44–66 (1995).

  22. M. K. Kwong and A. Zettl, “Norm inequalities for derivatives and differences,” Led. Notes Math., 1536 (1992).

  23. V. M. Tikhomirov and G. G. Magaril-Il’yacv, “Inequalities for derivatives,” in: Comments to Selected Works of A. N. Kolmogorov[in Russian], Nauka, Moscow (1985). pp. 387–390.

    Google Scholar 

  24. D. S. Mitrinovic, J. E. Pecharic, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer.Dordrecht (1991).

    MATH  Google Scholar 

  25. V. I. Burenkov, “On exact constants in inequalities for norms of intermediate derivatives on a Unite interval. I,” Tr. Mat. Inst. Akad.Nauk SSSR, 156, 22–29 (1980).

    MATH  MathSciNet  Google Scholar 

  26. V. I. Burenkov, “On exact constants in inequalities for norms of intermediate derivatives on a Unite interval. II,” Tr. Mat. Inst. Akad.Nauk SSSR, 173, 38–49 (1980).

    MathSciNet  Google Scholar 

  27. A. Yu. Shadrin, “To the Landau-Kolmogorov problem on a finite interval,” in: Proceedings of the International Conference “OpenProblems in Approximation Theom” (Voneshta Voda, June 18–24, 1993) (1993), pp. 192–204.

  28. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Additive inequalities for the intermediate derivatives of functions defined on afinite interval,” Ukr. Mat. Zh., 49, No. 5, 619–628 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  29. V. N. Gabushin, “Inequalities for norms of a function and its derivatives in the metrics of Lp,” Mat. Zametki, No. 3, 291–298 (1967).

  30. B. E. Klots, “Approximation of differentiable functions by functions of higher smoothness,” Mat. Zametki, 21, No. 1, 21–32 (1977).

    MATH  MathSciNet  Google Scholar 

  31. L. Hormander, “A new proof and generalization of inequality of Bohr,” Math. Scand., 2, 33–45 (1954).

    MathSciNet  Google Scholar 

  32. A. Sharma and T. Tzimbalario, “Landau-type inequalities for some linear differential operators,” III. J. Math., 20, No. 3, 443–455 (1976).

    MATH  MathSciNet  Google Scholar 

  33. Nguyen Thi Thieu Hoa, “On one extremal problem for classes of convolutions that do not increase oscillations,” Vesm. Most Univ.,Ser. Mat., Mekh., 5, 3–7 (1982).

    Google Scholar 

  34. Yongshen Syn, “Inequalities of Landau-Kolmogorov type for some linear differential operators,” Kexue Tongbao, 30, No. 8, 995–998 (1985).

    MathSciNet  Google Scholar 

  35. V. F. Babenko, “Extremal problems in approximation theory and inequalities for permutations,” Dokl. Akad. Nauk SSSR, 290, No. 5, 1033–1036 (1986).

    MathSciNet  Google Scholar 

  36. V. F. Babenko, Extremal Problems in Approximation Theory and Nonsymmetric Norms [in Russian], Doctoral-Degree Thesis(Physics and Mathematics), Kiev (1989).

  37. N. P. Korneichuk, “On the best uniform approximation on certain classes of continuous functions,” Dokl. Akad. Nauk SSSR, 140, No. 4, 748–751 (1961).

    MathSciNet  Google Scholar 

  38. N. P. Korneichuk, “Extremal values of functional and the best approximations on classes of periodic functions,” Izv. Akad. Nauk SSSR, Ser. Mat., 35, No. 1, 93–124 (1971).

    MathSciNet  Google Scholar 

  39. N. P. Korneichuk, “Inequalities for differentiable periodic functions and the best approximation of one class of functions by anotherclass,” Izv. Akad. Nauk SSSR, Ser. Mat., 36, No. 2, 423–434 (1972).

    MathSciNet  Google Scholar 

  40. N. P. Korneichuk, V. F. Babenko, V. A. Kofanov, and C. A. Pichugov, “Inequalities for upper bounds of functional and their applicationsto approximation theory,” Dopov. Akad. Nauk Ukr., No. 1, 24-29 (1999).

    Google Scholar 

  41. N. P. Korneichuk, V. F. Babenko, V. A. Kofanov, and C. A. Pichugov, “Inequalities for upper bounds of functionals on the classesW r H ω and their applications,” Ukr. Mat. Zh., 52, No. 1, 66–84 (2000).

    Article  MathSciNet  Google Scholar 

  42. A. Yu. Shadrin, “Kolmogorov-type inequalities and estimates of spline-interpolation for periodic functions from the class W m2 ,” Mat. Zametki, 48, No. 4, 132–139 (1990).

    MathSciNet  Google Scholar 

  43. M. W. Certain and T. G. Kurtz, “Landau-Kolmogorov inequalities for semigroups and groups,” Proc. Amer. Math. Soc, 63, 226–230 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  44. V. F. Babenko and S. A. Pichugov, “A comment to the Kolmogorov inequality,” in: Investigations on Contemporary Problems ofSummation and Approximation of Functions and Their Applications [in Russian], Dnepropetrovsk University, Dnepropetrovsk(1980), pp. 14–17.

    Google Scholar 

  45. V. F. Babenko and S. A. Pichugov, “On the Stein method,” in: Approximation of Functions and Summation of Series [in Russian), Dnepropetrovsk University, Dnepropetrovsk (1991), pp. 7–10.

    Google Scholar 

  46. A. A. Ligun, “Inequalities for upper bounds of functionals,” Analysis Math., 2, No. 1, 11–40 (1976).

    Article  MathSciNet  Google Scholar 

  47. V. F. Babenko, “Nonsymmetric extremal problems in approximation theory,” Dokl. Akad. Nauk SSSR, 269, No. 3, 521–524 (1983).

    MathSciNet  Google Scholar 

  48. V. F. Babenko, “Exact inequalities for norms of conjugate functions and their applications,” Ukr. Mat. Zh., 39, No. 2, 139–144 (1987).

    Article  MathSciNet  Google Scholar 

  49. V. F. Babenko, “Exact inequalities for norms of intermediate derivatives of half-integral order and their applications,” Dopov. Akad.Nauk Ukr., No. 2, 23–26 (1997).

    Google Scholar 

  50. V. F. Babenko, “Exact inequalities for norms of intermediate derivatives of half-integer order and some of their applications,” in:Approximation Theory and Applications, Hadronic Press, Palm Harbor (US A) (1998), pp. 5–16.

    Google Scholar 

  51. V. F. Babenko and V. N. Glushko, “On some exact inequalities for L,-norms of conjugate functions and their applications,” in:Contemporary Problems in Approximation Theory and Complex Analysis [in Russian], Institute of Mathematics, Ukrainian Academyof Sciences, Kiev (1990), pp. 5–12.

    Google Scholar 

  52. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “New exact inequalities of Kolmogorov type for periodic functions.” Dopov.Akad. Nauk Ukr., No. 7, 7–10 (1998).

  53. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Inequalities for norms of intermediate derivatives of periodic functions and theirapplications,” E. J. Approxim., 3, No. 3, 351–376 (1997).

    MATH  MathSciNet  Google Scholar 

  54. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Inequalities of Kolmogorov type and some of their applications in approximationtheory,” in: Proceedings of the Third International Conference on Functional Analysis and Approximation Theory(Acauafreda di Maratea (Potenza-Italy), September 23–28, 1996), Vol. I, Suppl. Rend. Cir. Mat. Palermo, Ser. II, No. 52 (1998),pp. 223–237.

  55. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “On the exact inequalities of Kolmogorov type and some of their applications,”in: New Approaches in Nonlinear Analysis, Hadronic Press, Palm Harbor (USA) (1999), pp. 9–50.

    Google Scholar 

  56. B. Sz. Nagy, “über Integralungleichungen zwischen einen Function und ihrer Ableitungee,” Acta Sci. Math., 10, 64–74 (1941).

    Google Scholar 

  57. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Sz.-Nagy-type inequalities for periodic functions,” in: Abstracts of the InternationalConference “Approximation Theory and Harmonic Analysis” (Tula, Russia, May 26–29, 1998) [in Russian], Tula (1998),p. 29.

  58. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Inequalities of the Sz.-Nagy type for periodic functions,” in: Proceedings ofthe International Mathematical Conference “Contemporary Problems in Mathematics” (Chernovtsy-Kiev) [in Russian], Part 1,(1998), pp. 9–11.

  59. V. F. Babenko, “Inequalities of Landau-Kolmogorov-Sz.-Nagy type,” in: Abstracts of the International Congress ofMathematicians (Berlin, August 18–27, 1998) (1998), p. 115.

  60. V. F. Babenko and M. B. Vakarchuk, “On inequalities of Kolmogorov-Hormander type for functions bounded on a discrete lattice,” Ukr. Mat. Zh., 49, No. 7, 988–992 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  61. V. N, Gabushin, “On some inequalities for derivatives of functions,” in: Methods for Regularization of Unstable Problems [in Russian],IMM UNTs AN SSSR, (1976), pp. 20–26.

  62. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “On exact inequalities of Kolmogorov type in the case of low smoothness,” Dopov. Akad. Nauk Ukr., No. 6, 11–15 (1998).

    Google Scholar 

  63. V. V. Arestov and V. I. Berdyshev, “Inequalities for differentiable functions,” in: Methods for the Solution of Conditionally Well-Posed Problems[in Russian], IMM UNTs AN SSSR, (1975), pp. 108–138.

  64. V. V. Arestov, “On exact inequalities for norms of functions and their derivatives,” Acta Sci. Math., 33, No. 3–4, 243–267 (1972).

    MATH  MathSciNet  Google Scholar 

  65. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “On some exact inequalities of Kolmogorov type for periodic functions,” in:Fourier Series: Theory and Applications [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev, (1998),pp. 30–42.

    Google Scholar 

  66. A. A. Ligun, “On inequalities for norms of derivatives of periodic functions,” Mat. Zametki, 33, No. 3, 385–391 (1983).

    MathSciNet  Google Scholar 

  67. V. F. Babenko and A. A. Ligun, “Bernstein-type inequalities for /.-splines,” Ukr. Mat. Zh., 45, No. 1, 10–20 (1993).

    Article  MathSciNet  Google Scholar 

  68. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “On exact inequalities of Kolmogorov type that take into account the number ofchanges of the sign of derivatives,” Dopov. Akad. Nauk Ukr., No. 8, 12–16 (1998).

    Google Scholar 

  69. S. B. Stechkin, “Inequalities for norms of derivatives of an arbitrary function,” Acta Sci. Math., 26. 225–230 (1965).

    MATH  Google Scholar 

  70. S. B. Stechkin, “The best approximation of linear operators,” Mat. Zametki, 1, No. 2, 137–148 (1967).

    MathSciNet  Google Scholar 

  71. L. V. Taikov, “On the best approximation in the mean for certain classes of analytic functions,” Mat. Zametki, 1, No. 2, 155–162 (1967).

    MathSciNet  Google Scholar 

  72. V. V. Arestov, “On certain extremal problems for differentiable functions of one variable,” Tr. Mat. Inst. Akad. Nauk SSSR, 138, 3–26 (1975).

    MATH  MathSciNet  Google Scholar 

  73. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Multivariate inequalities of Kolmogorov type and their applications,” in:G. Nurnbcrger, J. V. Schmidt, and G. Walz (eds.), Proceedings of the Mannheim Conference “Multivariate Approximation and Splines, 1996” (1997), pp. 1–12.

  74. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Kolmogorov-type inequalities for operators and extremal problems in approximationtheory,” Dokl. Ros. Akad. Nauk, 356, No. 4, 439–44 (1997).

    MathSciNet  Google Scholar 

  75. G. V. Milovanovic, D. S. Mitrinovic, and Th. M. Russias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, WorldScientific, Singapore 1994.

    MATH  Google Scholar 

  76. V. F. Babenko and A. A. Ligun, “Generalization of certain extremal properties of splines,” Ukr. Mat. Zh., 47, No. 3, 403–407 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  77. V. F. Babenko, “Inequalities for norms of intermediate derivatives and some their applications,” in: Recent Progress in Inequalities,Kluwer, Dordrecht (1998), pp. 77–96.

    Google Scholar 

  78. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “On inequalities for derivatives on a segment,” in: Approximation Theory andProblems in Computational Mathematics [in Russian ], Dnepropetrovsk (1993), p. 12.

  79. V. F. Babenko and Zh. B. Uedraogo, “On exact constants in inequalities for norms of derivatives on a finite segment,” Ukr. Mat. Zh., 51, No. 1, 117–119 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  80. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “On exact inequalities for intermediate derivatives of differentiable mappings ofBanach spaces,” Dopov. Akad. Nauk Ukr., No. 1, 22–25 (1997).

    Google Scholar 

  81. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “On additive inequalities for intermediate derivatives of differentiable mappingsof Banach spaces,” Mat. Zametki, 63, No. 3, 332–342 (1998).

    MathSciNet  Google Scholar 

  82. V. F. Babenko. V. A. Kofanov, and S. A. Pichugov, “On exact inequalities of Landau-Kolmogorov-Hormander type on a semiaxis,” Dopov. Akad. Nauk Ukr., No. 4, 34–38 (1997).

    Google Scholar 

  83. V. F. Babenko and S. A. Selivanova, “Relationship between Kolmogorov-type inequalities for periodic and nonperiodic functions,” in: Abstracts of the Second School “Fourier Series: Theory and Applications” (Kamenets-Podol’skii, June 30-July 5, 1997) [in Russian],Kiev (1997), pp. 19–20.

  84. V. F. Babenko and S. A. Selivanova, “On Kolmogorov-type inequalities for periodic and nonperiodic functions,” in: DifferentialEquations and Their Applications [in Russian], Dnepropetrovsk University, Dnepropetrovsk (1998), pp. 91–95.

    Google Scholar 

  85. V. F. Babenko and S. A. Selivanova, “On the connection between certain inequalities of the Kolmogorov type for periodic and non-periodic functions,” Ukr. Mat. Zh.. 51, No. 2, 147–157 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  86. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “On inequalities of Landau-Hadamard-Kolmogorov type for the L 2-norm of anintermediate derivative,” E. J. Approxim., 2, No. 3, 343–368 (1996).

    MATH  MathSciNet  Google Scholar 

  87. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “On exact inequalities of the Landau-Hadamard-Kolmogorov type for functionsof many variables,” Dokl. Ros. Akad. Nauk, 356, No. 1, 7–9 (1997).

    MathSciNet  Google Scholar 

  88. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Exact inequalities of Kolmogorov type for multivariate functions and their applications,” E. J. Approxim., 3, No. 2, 155–186 (1997).

    MATH  MathSciNet  Google Scholar 

  89. V. F. Babenko, “Exact inequalities of the Kolmogorov type and some of their applications,” in: Abstracts of the International Conferenceon Approximation Theory and its Applications Dedicated to the Memory of V. K. Dzyadyk. Kiev (1999), pp. 9–10.

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Babenko, V.F. Investigations of dnepropetrovsk mathematicians related to inequalities for derivatives of periodic functions and their applications. Ukr Math J 52, 8–28 (2000). https://doi.org/10.1007/BF02514133

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