On some extremal problems of different metrics for differentiable functions on the axis

  • V. A. Kofanov
Article
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For an arbitrary fixed segment [α, β] ⊂ R and given r ∈ N, A r , A 0, and p > 0, we solve the extremal problem
$$ \int\limits_\alpha^\beta {{{\left| {{x^{(k)}}(t)} \right|}^q}dt \to \sup, \,\,\,\,q \geqslant p,\,\,\,k = 0,\,\,\,q \geqslant 1,\,\,\,\,1 \leqslant k \leqslant r - 1,} $$
on the set of all functions x∈ L r ∞ such that ∥x (r)∥∞ ≤ A r and L(x) p ≤ A 0, where
$$ L{(x)_p}: = \left\{ {{{\left( {\int\limits_a^b {{{\left| {x(t)} \right|}^p}dt} } \right)}^{{1 \mathord{\left/{\vphantom {1 p}} \right.} p}}}:\,a,b \in R,\,\left| {x(t)} \right| > 0,\,t \in \left( {a,\,b} \right)} \right\} $$
In the case where p = ∞ and k ≥ 1, this problem was solved earlier by Bojanov and Naidenov.

Keywords

Arbitrary Function Naukova Dumka Differentiable Function Extremal Problem Comparison Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  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, “Investigation of Dnepropetrovsk mathematicians related to inequalities for derivatives of periodic functions and their applications,” Ukr. Mat. Zh., 52, No. 1, 9–29 (2000).MATHMathSciNetGoogle Scholar
  3. 3.
    M. K. Kwong and A. Zettl, Norm Inequalities for Derivatives and Differences, Springer, Berlin (1992).MATHGoogle 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).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    A. Pinkus and O. Shisha, “Variations on the Chebyshev and Lq theories of best approximation,” Approxim. Theory, 35, No. 2, 148–168 (1982).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    N. P. Korneichuk, V. F. Babenko, and A. A. Ligun, Extremal Properties of Polynomials and Splines [in Russian], Naukova Dumka, Kiev (1992).Google Scholar
  7. 7.
    A. N. Kolmogorov, “On inequalities between upper bounds of successive derivatives of a function on an infinite interval,” in: Selected Works, Mathematics and Mechanics [in Russian], Nauka, Moscow (1985), pp. 252–263.Google Scholar
  8. 8.
    A. A. Ligun, “Inequalities for upper bounds of functionals,” Anal. Math., 2, No. 1, 11–40 (1976).CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  • V. A. Kofanov
    • 1
  1. 1.Dnepropetrovsk National UniversityDnepropetrovskUkraine

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