Inequalities for upper bounds of functionals on the classes $W^r H^{ω}$ and their applications
We show that the well-known results on estimates of upper bounds of functionals on the classes $W^r H^{ω}$ of periodic functions can be regarded as a special case of Kolmogorov-type inequalities for support functions of convex sets. This enables us to prove numerous new statements concerning the app...
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| Date: | 2000 |
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| Main Authors: | , , , , , , , |
| Format: | Article |
| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2000
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4398 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510528556761088 |
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| author | Babenko, V. F. Korneichuk, N. P. Kofanov, V. A. Pichugov, S. A. Бабенко, В. Ф. Корнейчук, Н. П. Кофанов, В. А. Пичугов, С. А. Бабенко, В. Ф. Корнейчук, Н. П. Кофанов, В. А. Пичугов, С. А. |
| author_facet | Babenko, V. F. Korneichuk, N. P. Kofanov, V. A. Pichugov, S. A. Бабенко, В. Ф. Корнейчук, Н. П. Кофанов, В. А. Пичугов, С. А. Бабенко, В. Ф. Корнейчук, Н. П. 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 | 2020-03-18T20:28:25Z |
| description | We show that the well-known results on estimates of upper bounds of functionals on the classes $W^r H^{ω}$ of periodic functions can be regarded as a special case of Kolmogorov-type inequalities for support functions of convex sets. This enables us to prove numerous new statements concerning the approximation of the classes $W^r H^{ω}$, establish the equivalence of these statements, and obtain new exact inequalities of the Bernstein-Nikol’skii type that estimate the value of the support function of the class $H^{ω}$ on the derivatives of trigonometric polynomials or polynomial splines in terms of the $L^{ϱ}$ -norms of these polynomials and splines. |
| first_indexed | 2026-03-24T02:58:26Z |
| format | Article |
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| id | umjimathkievua-article-4398 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:58:26Z |
| publishDate | 2000 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/af/76cada398fc8ebe97e30fd0963a803af.pdf |
| spelling | umjimathkievua-article-43982020-03-18T20:28:25Z Inequalities for upper bounds of functionals on the classes $W^r H^{ω}$ and their applications О неравенствах для верхних граней функционалов на классах $W^r H^{ω}$ и некоторых их приложениях Babenko, V. F. Korneichuk, N. P. Kofanov, V. A. Pichugov, S. A. Бабенко, В. Ф. Корнейчук, Н. П. Кофанов, В. А. Пичугов, С. А. Бабенко, В. Ф. Корнейчук, Н. П. Кофанов, В. А. Пичугов, С. А. We show that the well-known results on estimates of upper bounds of functionals on the classes $W^r H^{ω}$ of periodic functions can be regarded as a special case of Kolmogorov-type inequalities for support functions of convex sets. This enables us to prove numerous new statements concerning the approximation of the classes $W^r H^{ω}$, establish the equivalence of these statements, and obtain new exact inequalities of the Bernstein-Nikol’skii type that estimate the value of the support function of the class $H^{ω}$ on the derivatives of trigonometric polynomials or polynomial splines in terms of the $L^{ϱ}$ -norms of these polynomials and splines. Показано, що відомі результати про оцінки верхніх граней функціоналів на класах $W^r H^{ω}$ періодичних функцій можна розглядати як спеціальний випадок нерівностей типу Колмогорова для опорних функцій опуклих, множин. Це дозволило одержати ряд нових тверджень, пов'язаних з апроксимацією класів $W^r H^{ω}$ та встановити їх еквівалентність, а також одержати нові точні нерівності типу Бернштейна-Нікольського, які оцінюють значення опорної функції класу $H^{ω}$ на похідних тригонометричних доліномів або поліношальних сплайнів через $L^{ϱ}$ -норми самих поліномів або сплайнів. Institute of Mathematics, NAS of Ukraine 2000-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4398 Ukrains’kyi Matematychnyi Zhurnal; Vol. 52 No. 1 (2000); 66-84 Український математичний журнал; Том 52 № 1 (2000); 66-84 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4398/5500 https://umj.imath.kiev.ua/index.php/umj/article/view/4398/5501 Copyright (c) 2000 Babenko V. F.; Korneichuk N. P.; Kofanov V. A.; Pichugov S. A. |
| spellingShingle | Babenko, V. F. Korneichuk, N. P. Kofanov, V. A. Pichugov, S. A. Бабенко, В. Ф. Корнейчук, Н. П. Кофанов, В. А. Пичугов, С. А. Бабенко, В. Ф. Корнейчук, Н. П. Кофанов, В. А. Пичугов, С. А. Inequalities for upper bounds of functionals on the classes $W^r H^{ω}$ and their applications |
| title | Inequalities for upper bounds of functionals on the classes $W^r H^{ω}$ and their applications |
| title_alt | О неравенствах для верхних граней функционалов на классах $W^r H^{ω}$ и некоторых их приложениях |
| title_full | Inequalities for upper bounds of functionals on the classes $W^r H^{ω}$ and their applications |
| title_fullStr | Inequalities for upper bounds of functionals on the classes $W^r H^{ω}$ and their applications |
| title_full_unstemmed | Inequalities for upper bounds of functionals on the classes $W^r H^{ω}$ and their applications |
| title_short | Inequalities for upper bounds of functionals on the classes $W^r H^{ω}$ and their applications |
| title_sort | inequalities for upper bounds of functionals on the classes $w^r h^{ω}$ and their applications |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4398 |
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