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Nonperiodic Sampling of Bandlimited Functions on Unions of Rectangular Lattices

Authors:	David Walnut

Institution:	(1) Department of Mathematical Sciences, George Mason University, Fairfax, Virginia 22030-4444, USA

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 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 A signal processing application in which such sampling sets arise naturally is described in detail.

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