On Kolmogorov-Type Inequalities with Integrable Highest Derivative
We obtain the new exact Kolmogorov-type inequality $$\left\| {x^{\left( k \right)} } \right\|_2 \leqslant K\left\| x \right\|_2^{\frac{{r - k - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}{{r - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace}...
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| Datum: | 2002 |
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| Hauptverfasser: | , , , , , |
| Format: | Artikel |
| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2002
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/4208 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510339707174912 |
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| author | Babenko, V. F. Kofanov, V. A. Pichugov, S. A. Бабенко, В. Ф. Кофанов, В. А. Пичугов, С. А. Бабенко, В. Ф. Кофанов, В. А. Пичугов, С. А. |
| author_facet | Babenko, V. F. Kofanov, V. A. Pichugov, S. A. Бабенко, В. Ф. Кофанов, В. А. Пичугов, С. А. Бабенко, В. Ф. Кофанов, В. А. Пичугов, С. А. |
| author_sort | Babenko, V. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:24:06Z |
| description | We obtain the new exact Kolmogorov-type inequality $$\left\| {x^{\left( k \right)} } \right\|_2 \leqslant K\left\| x \right\|_2^{\frac{{r - k - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}{{r - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}} \left\| {x^{\left( r \right)} } \right\|_1^{\frac{k}{{r{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}}$$ for 2π-periodic functions \(x \in L_1^r\) and any k, r ∈ N, k < r. We present applications of this inequality to problems of approximation of one class of functions by another class and estimates of K-functional type. |
| first_indexed | 2026-03-24T02:55:26Z |
| format | Article |
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| id | umjimathkievua-article-4208 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:55:26Z |
| publishDate | 2002 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/80/76cde0773ce0667a952efb43ca3a1680.pdf |
| spelling | umjimathkievua-article-42082020-03-18T20:24:06Z On Kolmogorov-Type Inequalities with Integrable Highest Derivative О неравенствах типа Колмогорова с интегрируемой старшей 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 the new exact Kolmogorov-type inequality $$\left\| {x^{\left( k \right)} } \right\|_2 \leqslant K\left\| x \right\|_2^{\frac{{r - k - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}{{r - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}} \left\| {x^{\left( r \right)} } \right\|_1^{\frac{k}{{r{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}}$$ for 2π-periodic functions \(x \in L_1^r\) and any k, r ∈ N, k < r. We present applications of this inequality to problems of approximation of one class of functions by another class and estimates of K-functional type. Одержано нову точну нерівність типу Колмогорова $$\left\| {x^{\left( k \right)} } \right\|_2 \leqslant K\left\| x \right\|_2^{\frac{{r - k - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}{{r - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}}} \left\| {x^{\left( r \right)} } \right\|_1^{\frac{k}{{r{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}}}$$ для $2π$-періодичних функцій $x \in L_1^r$ i довільних $k, r ∈ N,\; k < r$. Наведено застосування цієї нерівності в задачах наближення одного класу функцій іншим та оцінки типу $K$-функціонала. Institute of Mathematics, NAS of Ukraine 2002-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4208 Ukrains’kyi Matematychnyi Zhurnal; Vol. 54 No. 12 (2002); 1694-1697 Український математичний журнал; Том 54 № 12 (2002); 1694-1697 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4208/5123 https://umj.imath.kiev.ua/index.php/umj/article/view/4208/5124 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. On Kolmogorov-Type Inequalities with Integrable Highest Derivative |
| title | On Kolmogorov-Type Inequalities with Integrable Highest Derivative |
| title_alt | О неравенствах типа Колмогорова с интегрируемой старшей производной |
| title_full | On Kolmogorov-Type Inequalities with Integrable Highest Derivative |
| title_fullStr | On Kolmogorov-Type Inequalities with Integrable Highest Derivative |
| title_full_unstemmed | On Kolmogorov-Type Inequalities with Integrable Highest Derivative |
| title_short | On Kolmogorov-Type Inequalities with Integrable Highest Derivative |
| title_sort | on kolmogorov-type inequalities with integrable highest derivative |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4208 |
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