Kotel'nikov-type approximation theorems

We study approximation properties of the expansions $\sum_{k\in{\Bbb Z}^d}c_k\varphi(M^jx+k)$, where $M$ is a matrix dilation, $c_k$ is either the sampled value of a function $f$ at $M^{-j}k$ or its integral average near $M^{-j}k$ (falsified sampled value). Error estimations in $L_p$-norm, $2\le p\le\infty$, are given in terms of the Fourier transform of $f$. The approximation order depends on the decay of $\widehat f$ and on the order of Strang-Fix condition for $\phi$. The estimates are obtained for a wide class of $\varphi$ including both compactly supported and band-limited functions. The band-limited functions $\varphi$ provide an arbitrarily large approximation order, while the compactly supported functions are more preferable for implementations. For the one-dimensional case, we also constructed ‘’sampling wavelet decompositions’’, i.e. frame-like wavelet expansions with coefficients interpolating a function $f$ at the dyadic points.