Geodesic
On a method for interpolating functions on chaotic nets
Matematičeskie zametki, Tome 62 (1997) no. 3, pp. 404-417.

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Suppose $m,n\in\mathbb N$, $m\equiv n(\operatorname{mod}2)$, $K(x)=|x|^m$ for $m$ odd, $K(x)=|x|^m\ln|x|$ for $m$ even ($x\in\mathbb R^n$), $\mathscr P$ is the set of real polynomials in $n$ variables of total degree $\le m/2$, and $x_1,\dots,x_N\in \mathbb R^n$. We construct a function of the form $$ \sum_{j=1}^N\lambda_jK(x-x_j)+P(x), \qquad\text{where}\quad \lambda_j\in\mathbb R,\quad P\in\mathscr P,\quad \sum_{j=1}^N\lambda_jQ(x_j)=0\quad\forall Q\in\mathscr P, $$ coinciding with a given function $f(x)$ at the points $x_1,\dots,x_N$. Error estimates for the approximation of functions $f\in W_p^k(\Omega)$ and their $l$th-order derivatives in the norms $L_q(\Omega_\varepsilon)$ are obtained for this interpolation method, where $\Omega$ is a bounded domain in $\mathbb R^n$, $\varepsilon>0$, $\Omega_\varepsilon=\{x\in\Omega:\operatorname{dist}(x,\partial\Omega)>\varepsilon\}$. 
@article{MZM_1997_62_3_a8,
     author = {O. V. Matveev},
     title = {On a method for interpolating functions on chaotic nets},
     journal = {Matemati\v{c}eskie zametki},
     pages = {404--417},
     publisher = {mathdoc},
     volume = {62},
     number = {3},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_62_3_a8/}
}
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O. V. Matveev. On a method for interpolating functions on chaotic nets. Matematičeskie zametki, Tome 62 (1997) no. 3, pp. 404-417. http://geodesic.mathdoc.fr/item/MZM_1997_62_3_a8/

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