Kolmogorov-Type Inequalities for Periodic Functions Whose First Derivatives Have Bounded Variation
We obtain a new unimprovable Kolmogorov-type inequality for differentiable 2π-periodic functions x with bounded variation of the derivative x′, namely $$\left\| {x'} \right\|_q \leqslant K\left( {q,p} \right)\left\| x \right\|_p^a \left( {\mathop V\limits_{0}^{{2\pi }} \left( {x'...
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Institute of Mathematics, NAS of Ukraine
2002
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| author | Babenko, V. F. Kofanov, V. A. Pichugov, S. A. Бабенко, В. Ф. Кофанов, В. А. Пичугов, С. А. Бабенко, В. Ф. Кофанов, В. А. Пичугов, С. А. |
| author_facet | Babenko, V. F. Kofanov, V. A. Pichugov, S. A. Бабенко, В. Ф. Кофанов, В. А. Пичугов, С. А. Бабенко, В. Ф. Кофанов, В. А. Пичугов, С. А. |
| author_sort | Babenko, V. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:22:32Z |
| description | We obtain a new unimprovable Kolmogorov-type inequality for differentiable 2π-periodic functions x with bounded variation of the derivative x′, namely $$\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 }} ,$$ where q ∈ (0, ∞), p ∈ [1, ∞], and α = min{1/2, p/q(p + 1)}. |
| first_indexed | 2026-03-24T02:53:41Z |
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| id | umjimathkievua-article-4098 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | rus English |
| last_indexed | 2026-03-24T02:53:41Z |
| publishDate | 2002 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/93/37f5de87c6d5746fb9004a622f569b93.pdf |
| spelling | umjimathkievua-article-40982020-03-18T20:22:32Z Kolmogorov-Type Inequalities for Periodic Functions Whose First Derivatives Have Bounded Variation Неравенства типа Колмогорова для периодических функции с ограниченной вариацией первой производной Babenko, V. F. Kofanov, V. A. Pichugov, S. A. Бабенко, В. Ф. Кофанов, В. А. Пичугов, С. А. Бабенко, В. Ф. Кофанов, В. А. Пичугов, С. А. We obtain a new unimprovable Kolmogorov-type inequality for differentiable 2π-periodic functions x with bounded variation of the derivative x′, namely $$\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 }} ,$$ where q ∈ (0, ∞), p ∈ [1, ∞], and α = min{1/2, p/q(p + 1)}. Одержано нову непокращувану нерівність типу Колмогорова $$\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 }} ,$$ для дифереиційовних $2π$-періодичyих функцій $х$, що мають обмежену варіацію похідної $x′$, де $q ∈ (0, ∞), p ∈ [1, ∞]$, $α = min{1/2, p/q(p + 1)}$. Institute of Mathematics, NAS of Ukraine 2002-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4098 Ukrains’kyi Matematychnyi Zhurnal; Vol. 54 No. 5 (2002); 603-609 Український математичний журнал; Том 54 № 5 (2002); 603-609 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4098/4905 https://umj.imath.kiev.ua/index.php/umj/article/view/4098/4906 Copyright (c) 2002 Babenko V. F.; Kofanov V. A.; Pichugov S. A. |
| spellingShingle | Babenko, V. F. Kofanov, V. A. Pichugov, S. A. Бабенко, В. Ф. Кофанов, В. А. Пичугов, С. А. Бабенко, В. Ф. Кофанов, В. А. Пичугов, С. А. Kolmogorov-Type Inequalities for Periodic Functions Whose First Derivatives Have Bounded Variation |
| title | Kolmogorov-Type Inequalities for Periodic Functions Whose First Derivatives Have Bounded Variation |
| title_alt | Неравенства типа Колмогорова для периодических функции с ограниченной вариацией первой производной |
| title_full | Kolmogorov-Type Inequalities for Periodic Functions Whose First Derivatives Have Bounded Variation |
| title_fullStr | Kolmogorov-Type Inequalities for Periodic Functions Whose First Derivatives Have Bounded Variation |
| title_full_unstemmed | Kolmogorov-Type Inequalities for Periodic Functions Whose First Derivatives Have Bounded Variation |
| title_short | Kolmogorov-Type Inequalities for Periodic Functions Whose First Derivatives Have Bounded Variation |
| title_sort | kolmogorov-type inequalities for periodic functions whose first derivatives have bounded variation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4098 |
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