Simple quasicrystals are sets of stable sampling

Abstract

This contribution is addressing an issue named in signal processing. Let be a lattice and be the dual lattice. Then the standard Shannon–Nyquist theorem says that any signal f whose Fourier transform is supported by a compact subset can be recovered from the samples if and only if the translated sets are pairwise disjoint. This sufficient condition on K is also necessary. When it is not satisfied may occur. Olevskii and Ulanovskii designed irregular sampling strategies which remedy . Then one can optimally reduce the number of measures needed to recover a signal or an image whose Fourier transform is supported by a compact set K with a given measure. The present contribution is aimed at bridging the gap between this advance on irregular sampling and the theory of quasicrystals.