Nonperiodic Sampling of Bandlimited Functions on Unions of Rectangular Lattices

Abstract

It is shown that a function $f\in L^p[-R,R], 1\le p<\infty,$ is completely determined by the samples of $\hat f$ on sets $\Lambda=\cup_{i=1}^m\{n/2r_i\}_{n\in{\bf Z}}$ where $R=\sum r_i,$ and $r_i/r_j$ is irrational if $i\ne j,$ and of $\hat f^{(j)}(0) \mbox{ for } j=1,\ldots,m-1.$ If $f\in C^{m-2-k}[-R,R],$ then the samples of $\hat f$ on $\Lambda$ and only the first k derivatives of $\hat f$ at 0 are required to determine f completely. Higher dimensional analogues of these results, which apply to functions $f\in L^p[-R,R]^d$ and $C^{m-2-k}[-R,R]^d,$ are proven. The sampling results are sharp in the sense that if any condition is omitted, there exist nonzero $f\in L^p[-R,R]^d$ and $C^{m-2-k}[-R,R]^d$ satisfying the rest. It is shown that the one-dimensional sampling sets correspond to Bessel sequences of complex exponentials that are not Riesz bases for $L^2[-R,R].$ A signal processing application in which such sampling sets arise naturally is described in detail.