Inequality of the Bernshtein type for polynomial splines in the space $L_2$
For $2\pi$-periodic polynomial splines of order $r$, of minimal defect, with nodes at the points $k\pi /n, n\in\mathbb{N}$, there are established the sharp inequalities \[|| s^{(l)} || _{ 2} \leq \frac{|| \varphi _{n,r}^{(l)} || _{ 2} }{|| \Delta _h^l \varphi _{n, r} || _{ 2} }|| \Delta _h^l s||...
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| Date: | 1991 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Russian |
| Published: |
Institute of Mathematics, NAS of Ukraine
1991
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/9625 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860513544896774144 |
|---|---|
| author | Babenko , V. F. Pichugov , S. A. Бабенко , В. Ф. Пичугов , С. А. |
| author_facet | Babenko , V. F. Pichugov , S. A. Бабенко , В. Ф. Пичугов , С. А. |
| author_sort | Babenko , V. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-10-09T12:05:28Z |
| description | For $2\pi$-periodic polynomial splines of order $r$, of minimal defect, with nodes at the points $k\pi /n, n\in\mathbb{N}$, there are established the sharp inequalities
\[|| s^{(l)} || _{ 2} \leq \frac{|| \varphi _{n,r}^{(l)} || _{ 2} }{|| \Delta _h^l \varphi _{n, r} || _{ 2} }|| \Delta _h^l s|| _{ 2} \leq \frac{|| \varphi _{n,r}^{(l)} || _{ 2} }{|| \varphi _{n, r} || _{ 2} }|| s || _{ 2} ,\quad l = 1, \dots, r - 1,\]
valid for $0<h<\pi/2ln$ and $0<h<\pi/4ln$ respectively, where $\varphi_{n,r}$ is the $r$ -th periodic integral of the function $\varphi_{n,0}(x)={\rm sign} \, {\rm sin} nx$, and
$\Delta_h^lf(x)=\sum_{k=0}^l(-1)^kC_1^kf(x+(l-2k)h)$. |
| first_indexed | 2026-03-24T03:46:23Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-9625 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:46:23Z |
| publishDate | 1991 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/dc/1a90baae011fe7b90c94b99c51e30adc |
| spelling | umjimathkievua-article-96252025-10-09T12:05:28Z Inequality of the Bernshtein type for polynomial splines in the space $L_2$ Неравенства типа Бернштейна для полиномиальных сплайнов в пространстве $L_2$ Babenko , V. F. Pichugov , S. A. Бабенко , В. Ф. Пичугов , С. А. - For $2\pi$-periodic polynomial splines of order $r$, of minimal defect, with nodes at the points $k\pi /n, n\in\mathbb{N}$, there are established the sharp inequalities \[|| s^{(l)} || _{ 2} \leq \frac{|| \varphi _{n,r}^{(l)} || _{ 2} }{|| \Delta _h^l \varphi _{n, r} || _{ 2} }|| \Delta _h^l s|| _{ 2} \leq \frac{|| \varphi _{n,r}^{(l)} || _{ 2} }{|| \varphi _{n, r} || _{ 2} }|| s || _{ 2} ,\quad l = 1, \dots, r - 1,\] valid for $0<h<\pi/2ln$ and $0<h<\pi/4ln$ respectively, where $\varphi_{n,r}$ is the $r$ -th periodic integral of the function $\varphi_{n,0}(x)={\rm sign} \, {\rm sin} nx$, and $\Delta_h^lf(x)=\sum_{k=0}^l(-1)^kC_1^kf(x+(l-2k)h)$. Для $2\pi$-периодических полиномиальных сплайнов порядка $r$, минимального дефектна с узлами в точках $k\pi /n, n\in\mathbb{N}$ установлены точные неравенства \[|| s^{(l)} || _{ 2} \leq \frac{|| \varphi _{n,r}^{(l)} || _{ 2} }{|| \Delta _h^l \varphi _{n, r} || _{ 2} }|| \Delta _h^l s|| _{ 2} \leq \frac{|| \varphi _{n,r}^{(l)} || _{ 2} }{|| \varphi _{n, r} || _{ 2} }|| s || _{ 2} ,\quad l = 1, \dots, r - 1,\] справедливые соответственно при $0<h<\pi/2ln$, и $0<h<\pi/4ln$, где $\varphi_{n,r}$ — $r$-й периодический интеграл от функции $\varphi_{n,0}(x)={\rm sign} \, {\rm sin} nx$, а $\Delta_h^lf(x)=\sum_{k=0}^l(-1)^kC_1^kf(x+(l-2k)h)$. Для $2\pi$-періодичних поліпоміальних сплайнів порядку $r$, мінімального дефекту, з вузлами в точках $k\pi /n, n\in\mathbb{N}$, встановлені точні нерівності \[|| s^{(l)} || _{ 2} \leq \frac{|| \varphi _{n,r}^{(l)} || _{ 2} }{|| \Delta _h^l \varphi _{n, r} || _{ 2} }|| \Delta _h^l s|| _{ 2} \leq \frac{|| \varphi _{n,r}^{(l)} || _{ 2} }{|| \varphi _{n, r} || _{ 2} }|| s || _{ 2} ,\quad l = 1, \dots, r - 1,\] справедливі відповідно при $0<h<\pi/2ln$ і $0<h<\pi/4ln$, де $\varphi_{n,r}$ — $r$-й періодичний інтеграл від функції $\varphi_{n,0}(x)={\rm sign} \, {\rm sin} nx$, а $\Delta_h^lf(x)=\sum_{k=0}^l(-1)^kC_1^kf(x+(l-2k)h)$. Institute of Mathematics, NAS of Ukraine 1991-02-28 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/9625 Ukrains’kyi Matematychnyi Zhurnal; Vol. 43 No. 3 (1991); 420-422 Український математичний журнал; Том 43 № 3 (1991); 420-422 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/9625/10593 Copyright (c) 1991 В. Ф. Бабенко , С. А. Пичугов |
| spellingShingle | Babenko , V. F. Pichugov , S. A. Бабенко , В. Ф. Пичугов , С. А. Inequality of the Bernshtein type for polynomial splines in the space $L_2$ |
| title | Inequality of the Bernshtein type for polynomial splines in the space $L_2$ |
| title_alt | Неравенства типа Бернштейна для полиномиальных сплайнов в пространстве $L_2$ |
| title_full | Inequality of the Bernshtein type for polynomial splines in the space $L_2$ |
| title_fullStr | Inequality of the Bernshtein type for polynomial splines in the space $L_2$ |
| title_full_unstemmed | Inequality of the Bernshtein type for polynomial splines in the space $L_2$ |
| title_short | Inequality of the Bernshtein type for polynomial splines in the space $L_2$ |
| title_sort | inequality of the bernshtein type for polynomial splines in the space $l_2$ |
| topic_facet | - |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/9625 |
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