Multivariate exact and falsified sampling approximation

Abstract

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