Estimation of a K-Functional of Higher Order in Terms of a K-Functional of Lower Order
Let U j be a finite system of functionals of the form \(U_j (g):= \int _0^1 g^(k_j) ( \tau ) d \sigma _j ( \tau )+ \sum_{l < k_j} c_{j,l} g^(l) (0)\) , and let \(W_{p,U}^r\) be the subspace of the Sobolev space \(W_p^r [0;1]\) , 1 ≤ p ≤ +∞, that consists only of functions g such tha...
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Institute of Mathematics, NAS of Ukraine
2003
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| author | Radzievskaya, E. I. Radzievskii, G. V. Радзиевская, Е. И. Радзиевский, Г. В. Радзиевская, Е. И. Радзиевский, Г. В. |
| author_facet | Radzievskaya, E. I. Radzievskii, G. V. Радзиевская, Е. И. Радзиевский, Г. В. Радзиевская, Е. И. Радзиевский, Г. В. |
| author_sort | Radzievskaya, E. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:18:28Z |
| description | Let U j be a finite system of functionals of the form \(U_j (g):= \int _0^1 g^(k_j) ( \tau ) d \sigma _j ( \tau )+ \sum_{l < k_j} c_{j,l} g^(l) (0)\) , and let \(W_{p,U}^r\) be the subspace of the Sobolev space \(W_p^r [0;1]\) , 1 ≤ p ≤ +∞, that consists only of functions g such that U j(g) = 0 for k j < r. It is assumed that there exists at least one jump τ j for every function σ j , and if τ j = τ s for j ≠ s, then k j ≠ k s. For the K-functional $$K(\delta, f; L_p ,W_{p,U}^r) := \inf \limits_{g \in W_{p,U}^r} {|| f-g ||_p + \delta (|| g ||_p + || g^(r) ||_p)},$$ we establish the inequality \(K(\delta^n , f;L_p ,W_{p,U}^r) \leqslant cK(\delta^r ,f; L_p ,W_{p,U}^r)\) , where the constant c > 0 does not depend on δ ε (0; 1], the functions f belong to L p, and r = 1, ¨, n. On the basis of this inequality, we also obtain estimates for the K-functional in terms of the modulus of smoothness of a function f. |
| first_indexed | 2026-03-24T02:52:27Z |
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| id | umjimathkievua-article-4022 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | rus English |
| last_indexed | 2026-03-24T02:52:27Z |
| publishDate | 2003 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/60/9c0f4def068956d18e9f832579617a60.pdf |
| spelling | umjimathkievua-article-40222020-03-18T20:18:28Z Estimation of a K-Functional of Higher Order in Terms of a K-Functional of Lower Order Оценка K-функционала высокого порядка через K-функционал меньшего порядка Radzievskaya, E. I. Radzievskii, G. V. Радзиевская, Е. И. Радзиевский, Г. В. Радзиевская, Е. И. Радзиевский, Г. В. Let U j be a finite system of functionals of the form \(U_j (g):= \int _0^1 g^(k_j) ( \tau ) d \sigma _j ( \tau )+ \sum_{l < k_j} c_{j,l} g^(l) (0)\) , and let \(W_{p,U}^r\) be the subspace of the Sobolev space \(W_p^r [0;1]\) , 1 ≤ p ≤ +∞, that consists only of functions g such that U j(g) = 0 for k j < r. It is assumed that there exists at least one jump τ j for every function σ j , and if τ j = τ s for j ≠ s, then k j ≠ k s. For the K-functional $$K(\delta, f; L_p ,W_{p,U}^r) := \inf \limits_{g \in W_{p,U}^r} {|| f-g ||_p + \delta (|| g ||_p + || g^(r) ||_p)},$$ we establish the inequality \(K(\delta^n , f;L_p ,W_{p,U}^r) \leqslant cK(\delta^r ,f; L_p ,W_{p,U}^r)\) , where the constant c > 0 does not depend on δ ε (0; 1], the functions f belong to L p, and r = 1, ¨, n. On the basis of this inequality, we also obtain estimates for the K-functional in terms of the modulus of smoothness of a function f. Нехай $\{U_j\}$ — скінченна система функціоналів вигляду $$U_j (g):= \int _0^1 g^(k_j) ( \tau ) d \sigma _j ( \tau )+ \sum_{l < k_j} c_{j,l} g^(l) (0),$$ a $W_{p,U}^r$—підпростір простору Соболева $W_p^r [0;1], 1 ≤ p ≤ +∞,$ що складаєтьсяся лише з тих функцій $g$, для яких $U_j(g) = 0$ при $k_j < r$. Припускається, що для кожної функції $σ_j$ існує хоча б один стрибок $τj$, і якщо $τ_j = τ_s$ при $j ≠ s$, то $k_j ≠ k_s$. Для $K$-функціонала вигляду $$K(\delta, f; L_p ,W_{p,U}^r) := \inf \limits_{g \in W_{p,U}^r} {|| f-g ||_p + \delta (|| g ||_p + || g^(r) ||_p)},$$ встановлено нерівності$K(\delta^n , f;L_p ,W_{p,U}^r) \leqslant cK(\delta^r ,f; L_p ,W_{p,U}^r)$, де стала $c > 0 $ не залежить від $δ ε (0; 1]$, функції $f$ є $L_p$, і $r = 1, ¨, n.$ З цієї нерівності одержано також оцінки $К$-функціонала через модуль гладкості функції $f.$ Institute of Mathematics, NAS of Ukraine 2003-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4022 Ukrains’kyi Matematychnyi Zhurnal; Vol. 55 No. 11 (2003); 1530-1540 Український математичний журнал; Том 55 № 11 (2003); 1530-1540 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4022/4755 https://umj.imath.kiev.ua/index.php/umj/article/view/4022/4756 Copyright (c) 2003 Radzievskaya E. I.; Radzievskii G. V. |
| spellingShingle | Radzievskaya, E. I. Radzievskii, G. V. Радзиевская, Е. И. Радзиевский, Г. В. Радзиевская, Е. И. Радзиевский, Г. В. Estimation of a K-Functional of Higher Order in Terms of a K-Functional of Lower Order |
| title | Estimation of a K-Functional of Higher Order in Terms of a K-Functional of Lower Order |
| title_alt | Оценка K-функционала высокого порядка через K-функционал меньшего порядка |
| title_full | Estimation of a K-Functional of Higher Order in Terms of a K-Functional of Lower Order |
| title_fullStr | Estimation of a K-Functional of Higher Order in Terms of a K-Functional of Lower Order |
| title_full_unstemmed | Estimation of a K-Functional of Higher Order in Terms of a K-Functional of Lower Order |
| title_short | Estimation of a K-Functional of Higher Order in Terms of a K-Functional of Lower Order |
| title_sort | estimation of a k-functional of higher order in terms of a k-functional of lower order |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4022 |
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