Sampling of bandlimited functions on unions of shifted lattices (Q1602276)

The authors consider the Shannon sampling theory for (multidimensional, not necessarily periodic) sampling sets that are unions of shifted lattices. This theory is presented in the general framework of Fourier transforms on a locally compact abelian group \(G\). Then a continuous function \(f\in L^2(G)\) is reconstructed from its samples if the sampling set and the support of the Fourier transform of \(f\) satisfy certain compatibility conditions. An explicit reconstruction formula of \(f\) is given for sampling sets that are unions of 2 shifted lattices. Several examples are given, including a numerical example implemented in MATLAB. These methods provide a new tool for designing sampling sets of minimal density.