Regarding the Nyquist Criterion
The Nyquist criteria refers not to the frequency, but to the bandwidth, which is related to information density in a signal.  A very high frequency signal, of approximately known frequency, with a sufficiently small bandwidth, will still be aliased or folded down with baseband frequencies by undersampling.  But if the bandwidth (or other known characteristics) of the signal is known to rule out other aliased frequencies (such as the non-existence of baseband spectrum content), that knowledge in conjunction with the undersampled samples may still provide enough information to reconstruct the signal.