Estimation of the best approximation of periodic functions of two variables by an “angle” in the metric of $L_p$
We obtain upper bounds in terms of Fourier coefficients for the best approximation by an “angle” and for norms in the metric of $L_p$ for functions of two variables defined by trigonometric series with coefficients such that $a_{l_1 l_2} → 0$ as $l_1 + l_2 → ∞$ and $$\mathop \sum \limits_{k_1 = 0}^...
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| Дата: | 2004 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2004
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3834 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509966698283008 |
|---|---|
| author | Konopovich, T. O. Конопович, Т. О. |
| author_facet | Konopovich, T. O. Конопович, Т. О. |
| author_sort | Konopovich, T. O. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:12:11Z |
| description | We obtain upper bounds in terms of Fourier coefficients for the best approximation by an “angle” and for norms in the metric of $L_p$ for functions of two variables defined by trigonometric series with coefficients such that $a_{l_1 l_2} → 0$ as $l_1 + l_2 → ∞$ and
$$\mathop \sum \limits_{k_1 = 0}^\infty \mathop \sum \limits_{k_2 = 0}^\infty \left( {\mathop \sum \limits_{l_1 = k_1 }^\infty \mathop \sum \limits_{l_2 = k_2 }^\infty \left| {\Delta ^{12} a_{l_1 \;l_2 } } \right|} \right)^p (k_1 + 1)^{p - 2} \;(k_2 + 1)^{p - 2} < \infty$$
for a certain $p, 1 < p < ∞$. |
| first_indexed | 2026-03-24T02:49:30Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-3834 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:49:30Z |
| publishDate | 2004 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/84/5e102c6f5428780dc40ae80f72811084.pdf |
| spelling | umjimathkievua-article-38342020-03-18T20:12:11Z Estimation of the best approximation of periodic functions of two variables by an “angle” in the metric of $L_p$ Оцінка найкращого наближення „кутом" у мечриці $L_p$ періодичних функцій двох змінних Konopovich, T. O. Конопович, Т. О. We obtain upper bounds in terms of Fourier coefficients for the best approximation by an “angle” and for norms in the metric of $L_p$ for functions of two variables defined by trigonometric series with coefficients such that $a_{l_1 l_2} → 0$ as $l_1 + l_2 → ∞$ and $$\mathop \sum \limits_{k_1 = 0}^\infty \mathop \sum \limits_{k_2 = 0}^\infty \left( {\mathop \sum \limits_{l_1 = k_1 }^\infty \mathop \sum \limits_{l_2 = k_2 }^\infty \left| {\Delta ^{12} a_{l_1 \;l_2 } } \right|} \right)^p (k_1 + 1)^{p - 2} \;(k_2 + 1)^{p - 2} < \infty$$ for a certain $p, 1 < p < ∞$. Отримано ниражену через коефіцієнти Фур'є оцінку зверху найкращого наближення „кутом" та норми у метриці $L_p$ функцій двох змінних, які задані тригонометричними рядами з коефіцієнтами $a_{l_1 l_2} → 0$, $l_1 + l_2 → ∞$, що при деякому $p, 1 < p < ∞$ задовольняють умову $$\mathop \sum \limits_{k_1 = 0}^\infty \mathop \sum \limits_{k_2 = 0}^\infty \left( {\mathop \sum \limits_{l_1 = k_1 }^\infty \mathop \sum \limits_{l_2 = k_2 }^\infty \left| {\Delta ^{12} a_{l_1 \;l_2 } } \right|} \right)^p (k_1 + 1)^{p - 2} \;(k_2 + 1)^{p - 2} < \infty$$. Institute of Mathematics, NAS of Ukraine 2004-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3834 Ukrains’kyi Matematychnyi Zhurnal; Vol. 56 No. 9 (2004); 1182–1192 Український математичний журнал; Том 56 № 9 (2004); 1182–1192 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3834/4387 https://umj.imath.kiev.ua/index.php/umj/article/view/3834/4388 Copyright (c) 2004 Konopovich T. O. |
| spellingShingle | Konopovich, T. O. Конопович, Т. О. Estimation of the best approximation of periodic functions of two variables by an “angle” in the metric of $L_p$ |
| title | Estimation of the best approximation of periodic functions of two variables by an “angle” in the metric of $L_p$ |
| title_alt | Оцінка найкращого наближення „кутом" у мечриці $L_p$ періодичних функцій двох змінних |
| title_full | Estimation of the best approximation of periodic functions of two variables by an “angle” in the metric of $L_p$ |
| title_fullStr | Estimation of the best approximation of periodic functions of two variables by an “angle” in the metric of $L_p$ |
| title_full_unstemmed | Estimation of the best approximation of periodic functions of two variables by an “angle” in the metric of $L_p$ |
| title_short | Estimation of the best approximation of periodic functions of two variables by an “angle” in the metric of $L_p$ |
| title_sort | estimation of the best approximation of periodic functions of two variables by an “angle” in the metric of $l_p$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3834 |
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