Therefore, to some extent, compressed sensing is considered as a sampling technology similar to Shannon's sampling theorem
. However, the signal is sampled according to the sparsity of the signal but not the bandwidth required by Shannon's sampling theorem
It is worthwhile to highlight yet another aspect of our geometric sampling method: Shannon's Sampling theorem
relates to bandlimited signals, that are, necessarily, unbounded in time or in spatial domain.
The symmetries that underlie Shannon's sampling theorem
and its more general multi-band version are used as a basis for an exposition of sampling theory in a locally compact abelian group setting.