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 $$\frac{\partial ^s \bar O_{MN} f}{\partial v_k^s }\Bigg|_{\Gamma _k } = \frac{\partial ^s f}{\partial v_k^s }\Bigg|_{\Gamma _k } = {\varphi _{ks} (x)}\Bigg|_{\Gamma _...
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| Date: | 1992 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian |
| Published: |
Institute of Mathematics, NAS of Ukraine
1992
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/8251 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | A general algorithm is proposed for constructing interlineation $\bar O_{MN}f(x)$, $x = (x_1,x_2)$  with the properties
$$\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. |
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