2017
Том 69
№ 9

All Issues

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

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

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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)}.

English version (Springer): Ukrainian Mathematical Journal 54 (2002), no. 5, pp 741-749.

Citation Example: Babenko V. F., Kofanov V. A., Pichugov S. A. Kolmogorov-Type Inequalities for Periodic Functions Whose First Derivatives Have Bounded Variation // Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 603-609.

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