Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness
We obtain new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for r = 2, k = 1 or r = 3, k = 1, 2 and arbitrary q, p ∈ [1, ∞], the following unimprovable inequality holds for functions \(x \in L_\infty ^r \) : $$\left\| {x^{\left( k...
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| Дата: | 2001 |
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| Автори: | , , , , , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2001
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/4349 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510479773859840 |
|---|---|
| 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 | 2020-03-18T20:27:17Z |
| description | We obtain new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for r = 2, k = 1 or r = 3, k = 1, 2 and arbitrary q, p ∈ [1, ∞], the following unimprovable inequality holds for functions \(x \in L_\infty ^r \) : $$\left\| {x^{\left( k \right)} } \right\|_q \leqslant \frac{{\left\| {{\phi }_{r - k} } \right\|_q }}{{\left\| {{\phi }_r } \right\|_p^\alpha }}\left\| x \right\|_p^\alpha \left\| {x^{\left( k \right)} } \right\|_\infty ^{1 - \alpha } $$ where \(\alpha = \min \left\{ {1 - \frac{k}{r},\frac{{r - k + {1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-0em} q}}}{{r + {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-0em} p}}}} \right\}\) and ϕ r is the perfect Euler spline of order r. |
| first_indexed | 2026-03-24T02:57:39Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-4349 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:57:39Z |
| publishDate | 2001 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/52/e71a7a136cac0b8eaa716a297f0fcf52.pdf |
| spelling | umjimathkievua-article-43492020-03-18T20:27:17Z Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness Точные неравенства типа Колмогорова с ограниченной старшей производной в случае малых гладкостей 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 new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for r = 2, k = 1 or r = 3, k = 1, 2 and arbitrary q, p ∈ [1, ∞], the following unimprovable inequality holds for functions \(x \in L_\infty ^r \) : $$\left\| {x^{\left( k \right)} } \right\|_q \leqslant \frac{{\left\| {{\phi }_{r - k} } \right\|_q }}{{\left\| {{\phi }_r } \right\|_p^\alpha }}\left\| x \right\|_p^\alpha \left\| {x^{\left( k \right)} } \right\|_\infty ^{1 - \alpha } $$ where \(\alpha = \min \left\{ {1 - \frac{k}{r},\frac{{r - k + {1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-0em} q}}}{{r + {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-0em} p}}}} \right\}\) and ϕ r is the perfect Euler spline of order r. Одержано нові непокращувані нерівності типу Колмогорова для диференційовних періодичних функцій. Зокрема, доведено, що при $r = 2,\; k = 1$ або $r = 3,\; k = 1,\; 2$ та при довільних $q,p \in [1, \infty]$ для функцій $x \in L_{\infty}^r$, справедлива непокращувана нерівність $$\left\| {x^{\left( k \right)} } \right\|_q \leqslant \frac{{\left\| {{\phi }_{r - k} } \right\|_q }}{{\left\| {{\phi }_r } \right\|_p^\alpha }}\left\| x \right\|_p^\alpha \left\| {x^{\left( k \right)} } \right\|_\infty ^{1 - \alpha }$$ де $\alpha = \min \left\{ 1 - \frac kr, \frac{r - k + 1\backslash q}{r + 1 \backslash p} \right\}$ ($ϕ_r$— ідеальний сплайн Ейлера порядку $r$). Institute of Mathematics, NAS of Ukraine 2001-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4349 Ukrains’kyi Matematychnyi Zhurnal; Vol. 53 No. 10 (2001); 1299-1308 Український математичний журнал; Том 53 № 10 (2001); 1299-1308 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4349/5402 https://umj.imath.kiev.ua/index.php/umj/article/view/4349/5403 Copyright (c) 2001 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. Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness |
| title | Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness |
| title_alt | Точные неравенства типа Колмогорова с ограниченной старшей производной в случае малых гладкостей |
| title_full | Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness |
| title_fullStr | Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness |
| title_full_unstemmed | Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness |
| title_short | Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness |
| title_sort | exact kolmogorov-type inequalities with bounded leading derivative in the case of low smoothness |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4349 |
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