Interlination of the functions of 2 variables on $M (M\geq2)$ straight lines with the highest algebraic accuracy
A general algorithm is proposed for constructing interlineation $\bar O_{MN}f(x)$, $x = (x_1,x_2)$  with the properties $$\frac{\partial ^s \bar O_{MN} f}{\partial v_k^s }\Bigg|_{\Gamma _k } = \frac{\partial ^s f}{\partial v_k^s }\Bigg|_{\Gamma _k } = {\varphi _{ks} (x)}\Bigg|_{\Gamma _...
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| Date: | 1992 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian |
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Institute of Mathematics, NAS of Ukraine
1992
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/8251 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860513006957363200 |
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| author | Litvin, O.N. Литвин, А.Н. |
| author_facet | Litvin, O.N. Литвин, А.Н. |
| author_sort | Litvin, O.N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-03-27T09:55:08Z |
| description | A general algorithm is proposed for constructing interlineation $\bar O_{MN}f(x)$, $x = (x_1,x_2)$  with the properties
$$\frac{\partial ^s \bar O_{MN} f}{\partial v_k^s }\Bigg|_{\Gamma _k } = \frac{\partial ^s f}{\partial v_k^s }\Bigg|_{\Gamma _k } = {\varphi _{ks} (x)}\Bigg|_{\Gamma _k } ,k = \overline {1,M}; s = \overline {0,N} , $$
$$\bar O_{MN} x^\alpha \equiv x^\alpha ,0 \leq |\alpha | = \alpha _1 + \alpha _2 \leq M(N + 1) - 1, x^\alpha = x_1^{\alpha _1 } x_2^{\alpha _2 } ,$$
where ${\Gamma _k }$  is a given set of lines of arbitrary disposition on the plane $Ox_1x_2$, $v_k \bot \Gamma_k$. An integral representation is derived of the residual of approximation of the function $f(x)$ by the operators $\bar O_{MN} f(x)$. Examples are considered of interlineation operators preserving the class $C^r(R^2)$, and also operators not preserving the differentiability class, to which the function $f(x)$ belongs. |
| first_indexed | 2026-03-24T03:37:50Z |
| format | Article |
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| id | umjimathkievua-article-8251 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:37:50Z |
| publishDate | 1992 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/65/c73f61e5783b508f77d6273b4ed8ec65.pdf |
| spelling | umjimathkievua-article-82512024-03-27T09:55:08Z Interlination of the functions of 2 variables on $M (M\geq2)$ straight lines with the highest algebraic accuracy Интерлинация функций 2-х переменных на $M (M\geq2)$ прямых с наивысшей алгебраической точностью Litvin, O.N. Литвин, А.Н. A general algorithm is proposed for constructing interlineation $\bar O_{MN}f(x)$, $x = (x_1,x_2)$  with the properties $$\frac{\partial ^s \bar O_{MN} f}{\partial v_k^s }\Bigg|_{\Gamma _k } = \frac{\partial ^s f}{\partial v_k^s }\Bigg|_{\Gamma _k } = {\varphi _{ks} (x)}\Bigg|_{\Gamma _k } ,k = \overline {1,M}; s = \overline {0,N} , $$ $$\bar O_{MN} x^\alpha \equiv x^\alpha ,0 \leq |\alpha | = \alpha _1 + \alpha _2 \leq M(N + 1) - 1, x^\alpha = x_1^{\alpha _1 } x_2^{\alpha _2 } ,$$ where ${\Gamma _k }$  is a given set of lines of arbitrary disposition on the plane $Ox_1x_2$, $v_k \bot \Gamma_k$. An integral representation is derived of the residual of approximation of the function $f(x)$ by the operators $\bar O_{MN} f(x)$. Examples are considered of interlineation operators preserving the class $C^r(R^2)$, and also operators not preserving the differentiability class, to which the function $f(x)$ belongs. Предложен общий алгоритм построения операторов интерлинации $\bar O_{MN}f(x)$, $x = (x_1,x_2)$ со свойствами $$\frac{\partial ^s \bar O_{MN} f}{\partial v_k^s }\Bigg|_{\Gamma _k } = \frac{\partial ^s f}{\partial v_k^s }\Bigg|_{\Gamma _k } = {\varphi _{ks} (x)}\Bigg|_{\Gamma _k } ,k = \overline {1,M}; s = \overline {0,N} , $$ $$\bar O_{MN} x^\alpha \equiv x^\alpha ,0 \leq |\alpha | = \alpha _1 + \alpha _2 \leq M(N + 1) - 1, x^\alpha = x_1^{\alpha _1 } x_2^{\alpha _2 } ,$$ где ${\Gamma _k }$  — заданное множество прямых произвольного расположения на плоскости $Ox_1x_2$, $v_k \bot \Gamma_k$. Приведено интегральное представление остатка приближения функции $f(x)$ операторами $\bar O_{MN} f(x)$. Рассмотрены примеры операторов интерлинации с сохранением класса $C^r(R^2)$, а также операторов, не сохраняющих класс дифференцируемости, которому принадлежит функция $f(x)$. Запропоновано загальний алгоритм побудови операторів інтерлінації $\bar O_{MN}f(x)$, $x = (x_1,x_2)$ з властивостями $$\frac{\partial ^s \bar O_{MN} f}{\partial v_k^s }\Bigg|_{\Gamma _k } = \frac{\partial ^s f}{\partial v_k^s }\Bigg|_{\Gamma _k } = {\varphi _{ks} (x)}\Bigg|_{\Gamma _k } ,k = \overline {1,M}; s = \overline {0,N} , $$ $$\bar O_{MN} x^\alpha \equiv x^\alpha ,0 \leq |\alpha | = \alpha _1 + \alpha _2 \leq M(N + 1) - 1, x^\alpha = x_1^{\alpha _1 } x_2^{\alpha _2 } ,$$ де ${\Gamma _k }$ - задана множина прямих довільного розміщення на площині $Ox_1x_2$, $v_k \bot \Gamma_k$. Наведено інтегральне зображення залишку наближення функції $f(x)$ операторами $\bar O_{MN} f(x)$. Розглянуто приклади операторів інтерлінації із збереженням класу $C^r(R^2)$, а також операторів, які не зберігають клас диференційовності, якому належить функція $f(x)$. Institute of Mathematics, NAS of Ukraine 1992-11-06 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/8251 Ukrains’kyi Matematychnyi Zhurnal; Vol. 44 No. 11 (1992); 1498-1504 Український математичний журнал; Том 44 № 11 (1992); 1498-1504 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/8251/9831 Copyright (c) 1992 O.N. Litvin |
| spellingShingle | Litvin, O.N. Литвин, А.Н. Interlination of the functions of 2 variables on $M (M\geq2)$ straight lines with the highest algebraic accuracy |
| title | Interlination of the functions of 2 variables on $M (M\geq2)$ straight lines with the highest algebraic accuracy |
| title_alt | Интерлинация функций 2-х переменных на $M (M\geq2)$ прямых с наивысшей алгебраической точностью |
| title_full | Interlination of the functions of 2 variables on $M (M\geq2)$ straight lines with the highest algebraic accuracy |
| title_fullStr | Interlination of the functions of 2 variables on $M (M\geq2)$ straight lines with the highest algebraic accuracy |
| title_full_unstemmed | Interlination of the functions of 2 variables on $M (M\geq2)$ straight lines with the highest algebraic accuracy |
| title_short | Interlination of the functions of 2 variables on $M (M\geq2)$ straight lines with the highest algebraic accuracy |
| title_sort | interlination of the functions of 2 variables on $m (m\geq2)$ straight lines with the highest algebraic accuracy |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/8251 |
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