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