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On the Relationship between Sharp Kolmogorov-Type Inequalities and Sharp Kolmogorov–Remez-Type Inequalities

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Ukrainian Mathematical Journal Aims and scope

We establish a new theorem on the relationship between sharp constants in Kolmogorov-type inequalities and sharp constants in Kolmogorov–Remez-type inequalities for differentiable periodic functions. As a consequence, we obtain new sharp Kolmogorov–Remez-type inequalities for these functions. We also derive new sharp Bernstein–Remez-type inequalities for trigonometric polynomials and polynomial splines.

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References

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

    MathSciNet  MATH  Google Scholar 

  2. N. P. Korneichuk, V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, Inequalities for Derivatives and Their Applications [in Russian], Naukova Dumka, Kiev (2003).

    Google Scholar 

  3. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Exact Kolmogorov-type inequalities with bounded leading derivative in the case of low smoothness,” Ukr. Mat. Zh., 53, No. 10, 1298–1308 (2001); English translation: Ukr. Math. J., 53, No. 10, 1569–1582 (2001).

  4. V. F. Babenko, V. A. Kofanov, and S. A. Pichugov, “Comparison of exact constants in inequalities for derivatives of functions defined on the real axis and a circle,” Ukr. Mat. Zh., 55, No. 5, 579–589 (2003); English translation: Ukr. Math. J., 55, No. 5, 699–711 (2003).

  5. V. A. Kofanov and V. E. Mitropolskiy, “On the best constants in inequalities of Kolmogorov type,” East J. Approx., 13, No 4, 455–466 (2007).

    MathSciNet  Google Scholar 

  6. V. G. Solyar, “On one inequality for the norms of a function and its derivatives,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., 2, 165–168 (1976).

    Google Scholar 

  7. E. Remes, “Sur une propriete extremale des polynomes de Tchebychef,” Zap. Nauk.-Doslid. Inst. Mat. Mekh., Kharkiv. Mat. Tov., Ser. 4, 13, Issue 1, 93–95 (1936).

  8. M. I. Ganzburg, “On a Remez-type inequality for trigonometric polynomials,” J. Approx. Theory, 164, 1233–1237 (2012).

    Article  MathSciNet  Google Scholar 

  9. E. Nursultanov and S. Tikhonov, “A sharp Remez inequality for trigonometric polynomials,” Constr. Approx., 38, 101–132 (2013).

    Article  MathSciNet  Google Scholar 

  10. M. I. Ganzburg, “Polynomial inequalities on measurable sets and their applications,” Constr. Approx., 17, 275–306 (2001).

    Article  MathSciNet  Google Scholar 

  11. S. Tikhonov and P. Yuditski, Sharp Remez Inequality; https:www.researchgate.net/publication/327905401.

  12. V. A. Kofanov, “Sharp Remez-type inequalities for differentiable periodic functions, polynomials, and splines,” Ukr. Mat. Zh., 68, No. 2, 227–240 (2016); English translation: Ukr. Math. J., 68, No. 2, 253–268 (2016).

  13. B. Bojanov and N. Naidenov, “An extension of the Landau–Kolmogorov inequality. Solution of a problem of Erdos,” J. Anal. Math., 78, 263–280 (1999).

  14. V. A. Kofanov, “Sharp upper bounds of norms of functions and their derivatives on classes of functions with given comparison function,” Ukr. Mat. Zh., 63, No. 7, 969–984 (2011); English translation: Ukr. Math. J., 63, No. 7, 1118–1135 (2011).

  15. V. A. Kofanov, “Inequalities of different metrics for differentiable periodic functions,” Ukr. Mat. Zh., 67, No. 2, 202–212 (2015); English translation: Ukr. Math. J., 67, No. 2, 230–242 (2015).

  16. V. A. Kofanov, “Sharp Remez-type inequalities of different metrics for differentiable periodic functions, polynomials, and splines,” Ukr. Mat. Zh., 69, No. 2, 173–188 (2017); English translation: Ukr. Math. J., 69, No. 2, 205–223 (2017).

  17. A. E. Gaidabura and V. A. Kofanov, “Sharp Remez-type inequalities of various metrics in the classes of functions with given comparison function,” Ukr. Mat. Zh., 69, No. 11, 1472–1485 (2017); English translation: Ukr. Math. J., 69, No. 11, 1710–1726 (2018).

  18. V. A. Kofanov and I. V. Popovich, “Sharp Remez-type inequalities of various metrics with asymmetric restrictions imposed on the functions,” Ukr. Mat. Zh., 72, No. 7, 918–927 (2020); English translation: Ukr. Math. J., 72, No. 7, 1068–1079 (2020).

  19. V. A. Kofanov, “Sharp Kolmogorov–Remez-type inequalities for periodic functions of low smoothness,” Ukr. Mat. Zh., 72, No. 4, 483–493 (2020); English translation: Ukr. Math. J., 72, No. 4, 555–567 (2020).

  20. A. Pinkus and O. Shisha, “Variations on the Chebyshev and Lq theories of best approximation,” J. Approx. Theory, 35, No. 2, 148–168 (1982).

    Article  MathSciNet  Google Scholar 

  21. A. Zygmund, Trigonometric Series [Russian translation], Vol. 2, Mir, Moscow (1965).

    Google Scholar 

  22. V. V. Arestov, “On the integral inequalities for polynomials and splines,” Izv. Akad. Nauk SSSR, Ser. Mat., 45, 3–32 (1982).

    Google Scholar 

  23. V. A. Kofanov, “On exact Bernstein-type inequalities for splines,” Ukr. Mat. Zh., 58, No. 10, 1357–1367 (2006); English translation: Ukr. Math. J., 58, No. 10, 1538–1551 (2006).

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Authors and Affiliations

  1. Dnipro National University, Dnipro, Ukraine

    V. A. Kofanov

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  1. V. A. Kofanov
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Correspondence to V. A. Kofanov.

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Translated from Ukrains’kyi Matematychnyi Zhurnal,Vol. 73, No. 4, pp. 506–514, April, 2021. Ukrainian DOI: 10.37863/umzh.v73i4.6310.

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Kofanov, V.A. On the Relationship between Sharp Kolmogorov-Type Inequalities and Sharp Kolmogorov–Remez-Type Inequalities. Ukr Math J 73, 592–600 (2021). https://doi.org/10.1007/s11253-021-01945-8

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  • DOI: https://doi.org/10.1007/s11253-021-01945-8

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