Estimate of the Remainder of the Best Quadratic Approximation of Differentiable Functions by Polynomials
We establish lower and upper bounds for the quantity $$C_m^q (W^r ,x) = \mathop {\sup }\limits_{f \in W^r } \left| {f(x) - T_m (x,f)} \right|,$$ , where $$T_m (x,f) = \frac{2}{q}\mathop \sum \limits_{l = 0}^{q - 1} \;f(x_l )D_m (x - x_l ),\quad q \in \mathbb{N},\quad q > 2m,\quad x_l = \...
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| Дата: | 2004 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2004
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3876 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510007442800640 |
|---|---|
| author | Grigoryan, A. L. Григорян, А.Л. Григорян, А.Л. |
| author_facet | Grigoryan, A. L. Григорян, А.Л. Григорян, А.Л. |
| author_sort | Grigoryan, A. L. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:13:06Z |
| description | We establish lower and upper bounds for the quantity $$C_m^q (W^r ,x) = \mathop {\sup }\limits_{f \in W^r } \left| {f(x) - T_m (x,f)} \right|,$$ , where $$T_m (x,f) = \frac{2}{q}\mathop \sum \limits_{l = 0}^{q - 1} \;f(x_l )D_m (x - x_l ),\quad q \in \mathbb{N},\quad q > 2m,\quad x_l = \frac{{2\pi l}}{q},\quad l = 0,\;1,\;...\;,\;q - 1,$$ , and D m (t) is the Dirichlet kernel, for the class W r of 2π-periodic functions, whose rth derivative satisfies the condition |f r (x)| ≤ 1. |
| first_indexed | 2026-03-24T02:50:09Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-3876 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:50:09Z |
| publishDate | 2004 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/35/bc83db3ab67ff139f48c3820a7e17d35.pdf |
| spelling | umjimathkievua-article-38762020-03-18T20:13:06Z Estimate of the Remainder of the Best Quadratic Approximation of Differentiable Functions by Polynomials Оценка остатка наилучшего кпадратического приближения полиномами дифференцируемых функций Grigoryan, A. L. Григорян, А.Л. Григорян, А.Л. We establish lower and upper bounds for the quantity $$C_m^q (W^r ,x) = \mathop {\sup }\limits_{f \in W^r } \left| {f(x) - T_m (x,f)} \right|,$$ , where $$T_m (x,f) = \frac{2}{q}\mathop \sum \limits_{l = 0}^{q - 1} \;f(x_l )D_m (x - x_l ),\quad q \in \mathbb{N},\quad q > 2m,\quad x_l = \frac{{2\pi l}}{q},\quad l = 0,\;1,\;...\;,\;q - 1,$$ , and D m (t) is the Dirichlet kernel, for the class W r of 2π-periodic functions, whose rth derivative satisfies the condition |f r (x)| ≤ 1. Встановлено оцінки зверху і знизу величини $$C_m^q (W^r ,x) = \mathop {\sup }\limits_{f \in W^r } \left| {f(x) - T_m (x,f)} \right|,$$ де $$T_m (x,f) = \frac{2}{q}\mathop \sum \limits_{l = 0}^{q - 1} \;f(x_l )D_m (x - x_l ),$$ $$ q \in \mathbb{N},\quad q > 2m,\quad x_l = \frac{{2\pi l}}{q},\quad l = 0,\;1,\;...\;,\;q - 1,$$ $D_m(t)$ — ядро Діріхле, для класу $W^r$ $2π$-періодичпих функцій, що мають $r$-ту похідну, яка задовольняє умову $|f^r(x)| ≤ 1.$ Institute of Mathematics, NAS of Ukraine 2004-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3876 Ukrains’kyi Matematychnyi Zhurnal; Vol. 56 No. 12 (2004); 1691-1698 Український математичний журнал; Том 56 № 12 (2004); 1691-1698 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3876/4469 https://umj.imath.kiev.ua/index.php/umj/article/view/3876/4470 Copyright (c) 2004 Grigoryan A. L. |
| spellingShingle | Grigoryan, A. L. Григорян, А.Л. Григорян, А.Л. Estimate of the Remainder of the Best Quadratic Approximation of Differentiable Functions by Polynomials |
| title | Estimate of the Remainder of the Best Quadratic Approximation of Differentiable Functions by Polynomials |
| title_alt | Оценка остатка наилучшего кпадратического приближения полиномами дифференцируемых функций |
| title_full | Estimate of the Remainder of the Best Quadratic Approximation of Differentiable Functions by Polynomials |
| title_fullStr | Estimate of the Remainder of the Best Quadratic Approximation of Differentiable Functions by Polynomials |
| title_full_unstemmed | Estimate of the Remainder of the Best Quadratic Approximation of Differentiable Functions by Polynomials |
| title_short | Estimate of the Remainder of the Best Quadratic Approximation of Differentiable Functions by Polynomials |
| title_sort | estimate of the remainder of the best quadratic approximation of differentiable functions by polynomials |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3876 |
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