  The paper established sufficient conditions of predictability with degeneracy
for the spectrum at $M$-periodically located isolated points on the unit
circle. It is also shown that $m$-periodic subsequences of these sequences are
also predictable if $m$ is a divisor of $M$. The predictability can be achieved
for finite horizon with linear predictors defined by convolutions with certain
kernels. As an example of applications, it is shown that there exists a class
of sequences that is everywhere dense in the class of all square-summable
sequences and such that its members can be recovered from their periodic
subsequences. This recoverability is associated with certain spectrum
degeneracy of a new kind.
