  The paper investigates possibility of recovery of sequences from their
decimated subsequences. It is shown that this recoverability is associated with
certain spectrum degeneracy of a new kind, and that a sequences of a general
kind can be approximated by sequences featuring this degeneracy. This is
applied to sparse sampling of continuous time band-limited functions. The paper
shows that these functions allow an arbitrarily close approximation by
functions that can be recovered from sparse equidistant samples with sampling
distance larger than the distance defined by the critical Nyquist rate for the
underlying function. This allows to bypass, in a certain sense, the restriction
on the sampling rate defined by the Nyquist rate.
