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.