Moduli of continuity defined by zero continuation of functions and K-functionals with restrictions
We consider the followingK-functional: $$K(\delta ,f)_p : = \mathop {\sup }\limits_{g \in W_{p U}^r } \left\{ {\left\| {f - g} \right\|_{L_p } + \delta \sum\limits_{j = 0}^r {\left\| {g^{(j)} } \right\|_{L_p } } } \right\}, \delta \geqslant 0,$$ where ƒ ∈L p :=L p [0, 1] andW p,U r is a s...
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| Datum: | 1996 |
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| Format: | Artikel |
| Sprache: | Russisch Englisch |
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Institute of Mathematics, NAS of Ukraine
1996
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860511421981261824 |
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| author | Radzievskii, G. V. Радзиевский, Г. В. Радзиевский, Г. В. |
| author_facet | Radzievskii, G. V. Радзиевский, Г. В. Радзиевский, Г. В. |
| author_sort | Radzievskii, G. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:27:04Z |
| description | We consider the followingK-functional: $$K(\delta ,f)_p : = \mathop {\sup }\limits_{g \in W_{p U}^r } \left\{ {\left\| {f - g} \right\|_{L_p } + \delta \sum\limits_{j = 0}^r {\left\| {g^{(j)} } \right\|_{L_p } } } \right\}, \delta \geqslant 0,$$ where ƒ ∈L p :=L p [0, 1] andW p,U r is a subspace of the Sobolev spaceW p r [0, 1], 1≤p≤∞, which consists of functionsg such that \(\int_0^1 {g^{(l_j )} (\tau ) d\sigma _j (\tau ) = 0, j = 1, ... , n} \) . Assume that 0≤l l ≤...≤l n ≤r-1 and there is at least one point τ j of jump for each function σ j , and if τ j =τ s forj ≠s, thenl j ≠l s . Let \(\hat f(t) = f(t)\) , 0≤t≤1, let \(\hat f(t) = 0\) ,t0 does not depend on δ>0 and ƒ ∈L p . We also establish some other estimates for the consideredK-functional. |
| first_indexed | 2026-03-24T03:12:38Z |
| format | Article |
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| id | umjimathkievua-article-5208 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:12:38Z |
| publishDate | 1996 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/23/f378a4beba82ae6cb9aff186e1573523.pdf |
| spelling | umjimathkievua-article-52082020-03-18T21:27:04Z Moduli of continuity defined by zero continuation of functions and K-functionals with restrictions Модули непрерывности, определенные по нулевому продолжению функции, и $K$-функционалы с ограничениями Radzievskii, G. V. Радзиевский, Г. В. Радзиевский, Г. В. We consider the followingK-functional: $$K(\delta ,f)_p : = \mathop {\sup }\limits_{g \in W_{p U}^r } \left\{ {\left\| {f - g} \right\|_{L_p } + \delta \sum\limits_{j = 0}^r {\left\| {g^{(j)} } \right\|_{L_p } } } \right\}, \delta \geqslant 0,$$ where ƒ ∈L p :=L p [0, 1] andW p,U r is a subspace of the Sobolev spaceW p r [0, 1], 1≤p≤∞, which consists of functionsg such that \(\int_0^1 {g^{(l_j )} (\tau ) d\sigma _j (\tau ) = 0, j = 1, ... , n} \) . Assume that 0≤l l ≤...≤l n ≤r-1 and there is at least one point τ j of jump for each function σ j , and if τ j =τ s forj ≠s, thenl j ≠l s . Let \(\hat f(t) = f(t)\) , 0≤t≤1, let \(\hat f(t) = 0\) ,t0 does not depend on δ>0 and ƒ ∈L p . We also establish some other estimates for the consideredK-functional. Розглядається $K$-функціонал вигляду $$K(\delta ,f)_p : = \mathop {\sup }\limits_{g \in W_{p U}^r } \left\{ {\left\| {f - g} \right\|_{L_p } + \delta \sum\limits_{j = 0}^r {\left\| {g^{(j)} } \right\|_{L_p } } } \right\}, \delta \geqslant 0,$$ де $ƒ ∈ L_p := L_p [0, 1]$, a $W_p, U^r $ — підпростір простору Соболева $W_p^r [0, 1],\; 1 ≤ p ≤ ∞$, що складається з функцій $g$, для яких $\int_0^1 {g^{(l_j )} (\tau ) d\sigma _j (\tau ) = 0, j = 1, ... , n}$. Припускається, що $0 ≤ l_l ≤ ... ≤ l_n ≤ r-1$ та для кожної функції $τ_j$ існує хоча б один стрибок $σ_j$, якщо $τ_j = τ_s$, при $j ≠ s$, то $l_j ≠ l_s $. Для $l$ -го модуля неперервності функції $f$, заданого рівністю $$\hat \omega _0^{[l]} (\delta ,f)_p : = \mathop {\sup }\limits_{0 \leqslant h \leqslant \delta } \left\| {\sum\limits_{j = 0}^l {( - 1)^j \left( \begin{gathered} l \hfill \\ j \hfill \\ \end{gathered} \right)\hat f( - hj)} } \right\|_{L_p } , \delta \geqslant 0.$$ Де $\hat f(t) = f(t),\; 0 < t < 1$, і $\hat f(t) = 0, t < 0$, знайдено оцінки $ K(\delta ^r ,f)_p \leqslant c\hat \omega _0^{[l_1 ]} (\delta ,f)_p$, $K(\delta ^r ,f)_p \leqslant c\hat \omega _0^{[l_1 + 1]} (\delta ^\beta ,f)_p$ в яких $β=(pl_l + 1)/p(l_1 + 1)$, а стала $с > 0$ не залежить від $ δ>0$ і $ƒ ∈L_p$. Одержано також інші оцінки цього $K$-функціонала. Institute of Mathematics, NAS of Ukraine 1996-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5208 Ukrains’kyi Matematychnyi Zhurnal; Vol. 48 No. 11 (1996); 1537-1554 Український математичний журнал; Том 48 № 11 (1996); 1537-1554 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/5208/7110 https://umj.imath.kiev.ua/index.php/umj/article/view/5208/7111 Copyright (c) 1996 Radzievskii G. V. |
| spellingShingle | Radzievskii, G. V. Радзиевский, Г. В. Радзиевский, Г. В. Moduli of continuity defined by zero continuation of functions and K-functionals with restrictions |
| title | Moduli of continuity defined by zero continuation of functions and K-functionals with restrictions |
| title_alt | Модули непрерывности, определенные по нулевому продолжению
функции, и $K$-функционалы с ограничениями |
| title_full | Moduli of continuity defined by zero continuation of functions and K-functionals with restrictions |
| title_fullStr | Moduli of continuity defined by zero continuation of functions and K-functionals with restrictions |
| title_full_unstemmed | Moduli of continuity defined by zero continuation of functions and K-functionals with restrictions |
| title_short | Moduli of continuity defined by zero continuation of functions and K-functionals with restrictions |
| title_sort | moduli of continuity defined by zero continuation of functions and k-functionals with restrictions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5208 |
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