Some links while I have them at hand…
Compressed sensing basically gives a better practical way of recovering a signal from its samples than the Shannon Sampling theorem suggests – given some structure on the data.
Recent methods using L1 norm, as a compromise between L2 and L0, mean this is much more computationally feasible. New work by Tao, Candes etc gives some proofs on why this sparse sampling works so well.
Useful compressed sensing links -