On a method for interpolating functions on chaotic nets

O. V. Matveev

Matematičeskie zametki, Tome 62 (1997) no. 3, pp. 404-417

Résumé

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 $\mathbb R^n$, $\varepsilon>0$, $\Omega_\varepsilon=\{x\in\Omega:\operatorname{dist}(x,\partial\Omega)>\varepsilon\}$.