Interpolation by $D^m$-splines and bases in Sobolev spaces

Approximation of functions of several variables by $D^m$-interpolating splines on irregular grids is considered. Sharp in order estimates (of various kinds) of the error of the approximation of functions $f\in W^k_p(\Omega )$ in the seminorms ${\|D^l\cdot \|_{L_q}}$ are obtained in terms of the moduli of smoothness in $L_p$ of the $k$-th derivatives of $f$. As a consequence, for a bounded domain $\Omega$ in $\mathbb R^n$ with minimally smooth boundary and for each $t\in \mathbb N$ a basis in the Sobolev space $W^k_p(\Omega )$ is constructed such that the error of the approximation of $f\in W^k_p(\Omega )$ by the $N$-th partial sum of the expansion of $f$ with respect to this basis has an estimate in terms of its $t$-th modulus of smoothness $\omega _t(D^kf,N^{-1/n})_{L_p(\Omega )}$.