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Ukrainian Mathematical Journal

, Volume 54, Issue 5, pp 741–749 | Cite as

Kolmogorov-Type Inequalities for Periodic Functions Whose First Derivatives Have Bounded Variation

  • V. F. Babenko
  • V. A. Kofanov
  • S. A. Pichugov
Article
  • 24 Downloads

Abstract

We obtain a new unimprovable Kolmogorov-type inequality for differentiable 2π-periodic functions x with bounded variation of the derivative x′, namely
$$\left\| {x'} \right\|_q \leqslant K\left( {q,p} \right)\left\| x \right\|_p^a \left( {\mathop V\limits_{0}^{{2\pi }} \left( {x'} \right)} \right)^{1 - {alpha }} ,$$
where q ∈ (0, ∞), p ∈ [1, ∞], and α = min{1/2, p/q(p + 1)}.

Keywords

Periodic Function Bound Variation 
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Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • V. F. Babenko
    • 1
  • V. A. Kofanov
    • 1
  • S. A. Pichugov
    • 1
  1. 1.Dnepropetrovsk UniversityDnepropetrovsk

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