  Donoho and Stark have shown that a precise deterministic recovery of missing
information contained in a time interval shorter than the time-frequency
uncertainty limit is possible. We analyze this signal recovery mechanism from a
physics point of view and show that the well-known Shannon-Nyquist sampling
theorem, which is fundamental in signal processing, also uses essentially the
same mechanism. The uncertainty relation in the context of information theory,
which is based on Fourier analysis, provides a criterion to distinguish
Shannon-Nyquist sampling from compressed sensing. A new signal recovery
formula, which is analogous to Donoho-Stark formula, is given using the idea of
Shannon-Nyquist sampling; in this formulation, the smearing of information
below the uncertainty limit as well as the recovery of information with
specified bandwidth take place. We also discuss the recovery of states from the
domain below the uncertainty limit of coordinate and momentum in quantum
mechanics and show that in principle the state recovery works by assuming ideal
measurement procedures. The recovery of the lost information in the
sub-uncertainty domain means that the loss of information in such a small
domain is not fatal, which is in accord with our common understanding of the
uncertainty principle, although its precise recovery is something we are not
used to in quantum mechanics. The uncertainty principle provides a universal
sampling criterion covering both the classical Shannon-Nyquist sampling theorem
and the quantum mechanical measurement.
