Ukrainian Mathematical Journal

, Volume 71, Issue 6, pp 896–911 | Cite as

Bojanov–Naidenov Problem for Differentiable Functions on the Real Line and the Inequalities of Various Metrics

  • V. A. KofanovEmail author

For given r ∈ N, p, λ > 0 and any fixed interval [a, b] R, we solve the extreme problem

$$ \underset{a}{\overset{b}{\int }}{\left|x(t)\right|}^q dt\to \sup, \kern0.5em q\ge p, $$
on a set of functions \( x\in {L}_{\infty}^r \) such that
$$ {\displaystyle \begin{array}{ccc}{\left\Vert {x}^{(r)}\right\Vert}_{\infty}\le 1,& {\left\Vert x\right\Vert}_{p,\updelta}\le {\left\Vert {\upvarphi \uplambda}_{,r}\right\Vert}_{p,\updelta, }& \delta \in \Big(0,\uppi /\uplambda \end{array}}, $$


\( {\displaystyle \begin{array}{cc}{\left\Vert x\right\Vert}_{p,\delta}:= \sup \left\{\left\Vert x\right\Vert {L}_{p\left[a,b\right]}:a,b\in \mathbf{R},\right.& \left.0<b-a\le \updelta \right\}\end{array}} \)

and φ⋌,r is a (2𝜋/λ)-periodic Euler spline of order r. In particular, we solve the same problem for the intermediate derivatives x(k), k=1, . . . , r−1, with q≥1. In addition, we prove the inequalities of various metrics for the quantities ||x||p,𝜹.


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  1. 1.
    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
  2. 2.
    V. F. Babenko, “Investigations of Dnepropetrovsk mathematicians related to inequalities for derivatives of periodic functions and their applications,” Ukr. Mat. Zh., 52, No. 1, 5–29 (2000); English translation: Ukr. Math. J., 52, No. 1, 8–28 (2000).CrossRefGoogle Scholar
  3. 3.
    M. K. Kwong and A. Zettl, Norm Inequalities for Derivatives and Differences, Springer, Berlin (1992).CrossRefGoogle Scholar
  4. 4.
    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).MathSciNetCrossRefGoogle Scholar
  5. 5.
    A. Pinkus and O. Shisha, “Variations on the Chebyshev and Lq theories of best approximation,” J. Approxim. Theory, 35, No. 2, 148–168 (1982).CrossRefGoogle Scholar
  6. 6.
    V. A. Kofanov, “On some extremal problems of different metrics for differentiable functions on the axis,” Ukr. Mat. Zh., 61, No. 6, 765–776 (2009); English translation: Ukr. Math. J., 61, No. 6, 908–922 (2009).MathSciNetCrossRefGoogle Scholar
  7. 7.
    V. A. Kofanov, “Some extremal problems various metrics and sharp inequalities of Nagy–Kolmogorov type,” East J. Approxim., 16, No 4, 313–334 (2010).MathSciNetzbMATHGoogle Scholar
  8. 8.
    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).MathSciNetCrossRefGoogle Scholar
  9. 9.
    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).MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. N. Kolmogorov, “On the inequalities between upper bounds of successive derivatives on an infinite interval,” in: Selected Works, Mathematics and Mechanics [in Russian], Nauka, Moscow (1985), pp. 252–263.Google Scholar
  11. 11.
    N. P. Korneichuk, V. F. Babenko, and A. A. Ligun, Extremal Properties of Polynomials and Splines [in Russian], Naukova Dumka, Kiev (1992).zbMATHGoogle Scholar
  12. 12.
    V. A. Kofanov, “Sharp inequalities of Bernstein and Kolmogorov type,” East. J. Approxim., 11, No. 2, 131–145 (2005).MathSciNetzbMATHGoogle Scholar
  13. 13.
    N. P. Korneichuk, A. A. Ligun, and V. G. Doronin, Approximations with Restrictions [in Russian], Naukova Dumka, Kiev (1982).Google Scholar
  14. 14.
    B. M. Levitan, Almost Periodic Functions [in Russian], Gostekhizdat, Moscow (1953).Google Scholar
  15. 15.
    H. Weyl, “Almost periodic invariant vector sets in a metric vector space,” Amer. J. Math., 71, No. 1, 178–205 (1949).MathSciNetCrossRefGoogle Scholar
  16. 16.
    V. F. Babenko and S. A. Selivanov, “On the Kolmogorov-type inequalities for periodic and nonperiodic functions,” in: Differential Equations and Their Applications [in Russian], Dnipropetrovs’k National University, Dnipropetrovs’k (1998), pp. 91–95.Google Scholar
  17. 17.
    V. A. Kofanov, “Inequalities for nonperiodic splines on the real axis and their derivatives,” Ukr. Mat. Zh., 66, No. 2, 216–225 (2014); English translation: Ukr. Math. J., 66, No. 2, 242–252 (2014).MathSciNetCrossRefGoogle Scholar

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

Authors and Affiliations

  1. 1.Dnepr National UniversityDniproUkraine

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