Skip to main content
SpringerLink
Log in

Bojanov–Naidenov Problem for Differentiable Functions and the Erdős Problem for Polynomials and Splines

  • Published:
Ukrainian Mathematical Journal Aims and scope

We solve an extremal problem

$${\Vert {x}_{\pm }^{\left(k\right)}\Vert }_{{L}_{p}\left[a,b\right]}\to \mathrm{sup}, k=\mathrm{0,1},\dots ,\mathrm{r}-1,\mathrm{p}>0,$$

in a class of pairs (x, I) of functions \(x \in {S}_{\varphi }^{k}\) such that \({\varphi }^{\left(i\right)}\) are the comparison functions for \({x}^{\left(i\right)},\) i = 0, 1,…,k, and the intervals I = [a, b] satisfy the conditions

$${L\left(x\right)}_{p}\le A, \mu \left\{{\mathrm{supp}}_{\left[a,b\right]}{x}_{\pm }^{\left(k\right)}\right\}\le \mu ,$$

where

$${L\left(x\right)}_{p}:=\mathrm{sup}\left\{{\left(\underset{a}{\overset{b}{\int }}{\left|x\left(t\right)\right|}^{p}dt\right)}^\frac{1}{p}:a,b\in \mathbf{R},\left|x\left(t\right)\right|>0,t\in \left(a,b\right)\right\}.$$

In particular, we solve the same problems on the classes \({W}_{\infty }^{r}\left(\mathbf{R}\right)\) and on bounded sets of spaces of trigonometric polynomials and splines, as well as the Erdős problem for the positive (negative) parts of polynomials and splines.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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. 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).

  3. M. K. Kwong and A. Zettl, Norm Inequalities for Derivatives and Differences, Springer, Berlin (1992).

    Book  MATH  Google Scholar 

  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. 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).

    Article  MathSciNet  MATH  Google Scholar 

  6. P. Erd˝os, “Open problems,” in: B. Bojanov (editor), Open Problems in Approximation Theory, SCT Publ., Singapore (1994), pp. 238–242.

  7. 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).

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

    Article  MATH  Google Scholar 

  9. 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).

  10. V. A. Kofanov, “Some extremal problems of various metrics and sharp inequalities of Nagy–Kolmogorov type,” East J. Approxim., 16, No 4, 313–334 (2010).

    MathSciNet  MATH  Google Scholar 

  11. 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).

  12. V. A. Kofanov, “Bojanov–Naidenov problem for functions with asymmetric restrictions for the higher derivative,” Ukr. Mat. Zh., 71, No. 3, 368–381 (2019); English translation: Ukr. Math. J., 71, No. 3, 419–434 (2019).

  13. V. A. Kofanov, “Inequalities for derivatives of functions on an axis with nonsymmetrically bounded higher derivatives,” Ukr. Mat. Zh., 64, No 5, 636–648 (2012); English translation: Ukr. Math. J., 64, No 5, 721–736 (2012).

  14. V. A. Kofanov, “Bojanov–Naidenov problem for differentiable functions on the real line and the inequalities of various metrics,” Ukr. Mat. Zh., 71, No. 6, 786–800 (2019); English translation: Ukr. Math. J., 71, No. 6, 896–911 (2019).

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

    MATH  Google Scholar 

  17. V. V. Kameneva and V. A. Kofanov, “Bojanov–Naidenov problem for positive (negative) parts of differentiable functions on the real domain,” Res. Math., 26, 25–30 (2018).

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Volodymyr Kofanov

Authors
  1. Volodymyr Kofanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Volodymyr Kofanov.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, No. 2, pp. 182–197, February, 2023. Ukrainian DOI: https://doi.org/10.37863/umzh.v75i2.7259.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kofanov, V. Bojanov–Naidenov Problem for Differentiable Functions and the Erdős Problem for Polynomials and Splines. Ukr Math J 75, 206–224 (2023). https://doi.org/10.1007/s11253-023-02194-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-023-02194-7

Search

Navigation