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

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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)}.

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Authors and Affiliations

  1. Dnepropetrovsk University, Dnepropetrovsk

    V. F. Babenko, V. A. Kofanov & S. A. Pichugov

Authors
  1. V. F. Babenko
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  2. V. A. Kofanov
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  3. S. A. Pichugov
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Babenko, V.F., Kofanov, V.A. & Pichugov, S.A. Kolmogorov-Type Inequalities for Periodic Functions Whose First Derivatives Have Bounded Variation. Ukrainian Mathematical Journal 54, 741–749 (2002). https://doi.org/10.1023/A:1021675111923

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