Exact constants in inequalities of the jackson type for quadrature formulas
We prove that if \(R_n \left( {f,\{ t_k \} ,\{ p_k \} } \right)\) is the error of a simple quadrature formula and ω(ε, δ)1 is the integral modulus of continuity, then, for any δ ≥/π andn,r = 1, 2, …, the following equality is true: \(\mathop {\inf }\limits_{\{ f_k \} ,\{ p_k \} } \mathop {\sup }...
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| Date: | 2000 |
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| 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/4395 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510524880453632 |
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| author | Doronin, V. G. Ligun, A. A. Доронин, В. Г. Лигун, А. А. Доронин, В. Г. Лигун, А. А. |
| author_facet | Doronin, V. G. Ligun, A. A. Доронин, В. Г. Лигун, А. А. Доронин, В. Г. Лигун, А. А. |
| author_sort | Doronin, V. G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:28:25Z |
| description | We prove that if \(R_n \left( {f,\{ t_k \} ,\{ p_k \} } \right)\) is the error of a simple quadrature formula and ω(ε, δ)1 is the integral modulus of continuity, then, for any δ ≥/π andn,r = 1, 2, …, the following equality is true: \(\mathop {\inf }\limits_{\{ f_k \} ,\{ p_k \} } \mathop {\sup }\limits_{f \in L_1^r \backslash R_1 } \frac{{\left| {R_n (f,\{ t_k \} ,\{ p_k \} )} \right|}}{{\omega (f^{(r)} ,\delta )_1 }} = \frac{{\pi \left\| {D_1 } \right\|_\infty }}{{n^r }}\) whereD r is the Bernoulli kernel. |
| first_indexed | 2026-03-24T02:58:22Z |
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| id | umjimathkievua-article-4395 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:58:22Z |
| publishDate | 2000 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/4d/016d43e6662c7fa295a9bec719f6ce4d.pdf |
| spelling | umjimathkievua-article-43952020-03-18T20:28:25Z Exact constants in inequalities of the jackson type for quadrature formulas Точные константы в неравенствах типа Джексона для квадратурных формул Doronin, V. G. Ligun, A. A. Доронин, В. Г. Лигун, А. А. Доронин, В. Г. Лигун, А. А. We prove that if \(R_n \left( {f,\{ t_k \} ,\{ p_k \} } \right)\) is the error of a simple quadrature formula and ω(ε, δ)1 is the integral modulus of continuity, then, for any δ ≥/π andn,r = 1, 2, …, the following equality is true: \(\mathop {\inf }\limits_{\{ f_k \} ,\{ p_k \} } \mathop {\sup }\limits_{f \in L_1^r \backslash R_1 } \frac{{\left| {R_n (f,\{ t_k \} ,\{ p_k \} )} \right|}}{{\omega (f^{(r)} ,\delta )_1 }} = \frac{{\pi \left\| {D_1 } \right\|_\infty }}{{n^r }}\) whereD r is the Bernoulli kernel. Доведено, що якщо $R_n \left( {f,\{ t_k \} ,\{ p_k \} } \right)$ — похибка простої квадратурної формули та $ω(ε, δ)_1$ — інтегральний модуль неперервності, то для довільних $δ ≥/π$ при будь-яких $n, r = 1, 2, …,$ справджується рівність $$\mathop {\inf }\limits_{\{ f_k \} ,\{ p_k \} } \mathop {\sup }\limits_{f \in L_1^r \backslash R_1 } \frac{{\left| {R_n (f,\{ t_k \} ,\{ p_k \} )} \right|}}{{\omega (f^{(r)} ,\delta )_1 }} = \frac{{\pi \left\| {D_1 } \right\|_\infty }}{{n^r }}$$ де $D_r $— ядро Бернуллі. Institute of Mathematics, NAS of Ukraine 2000-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4395 Ukrains’kyi Matematychnyi Zhurnal; Vol. 52 No. 1 (2000); 46-51 Український математичний журнал; Том 52 № 1 (2000); 46-51 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4395/5494 https://umj.imath.kiev.ua/index.php/umj/article/view/4395/5495 Copyright (c) 2000 Doronin V. G.; Ligun A. A. |
| spellingShingle | Doronin, V. G. Ligun, A. A. Доронин, В. Г. Лигун, А. А. Доронин, В. Г. Лигун, А. А. Exact constants in inequalities of the jackson type for quadrature formulas |
| title | Exact constants in inequalities of the jackson type for quadrature formulas |
| title_alt | Точные константы в неравенствах типа Джексона для
квадратурных формул |
| title_full | Exact constants in inequalities of the jackson type for quadrature formulas |
| title_fullStr | Exact constants in inequalities of the jackson type for quadrature formulas |
| title_full_unstemmed | Exact constants in inequalities of the jackson type for quadrature formulas |
| title_short | Exact constants in inequalities of the jackson type for quadrature formulas |
| title_sort | exact constants in inequalities of the jackson type for quadrature formulas |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4395 |
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