Multivariate exact and falsified sampling approximation

摘要: Approximation properties of the expansions $\sum_{k\in{\mathbb z}^d}c_k\phi(M^jx+k)$, where $M$ is a matrix dilation, $c_k$ either sampled value signal $f$ at $M^{-j}k$ or integral average near (falsified value), are studied. Error estimations in $L_p$-norm, $2\le p\le\infty$, given terms Fourier transform $f$. The approximation order depends on how smooth $f$, Strang-Fix condition for $\phi$ and $M$. Some special required. To estimate falsified sampling we compare them with differential $\sum_{k\in\,{\mathbb z}^d} Lf(M^{-j}\cdot)(-k)\phi(M^jx+k)$, $L$ an appropriate operator. concrete functions applicable implementations constructed. In particular, compactly supported splines band-limited can be taken as $\phi$. these provide interpolating points $M^{-j}k$.