Predictability of sequences and subsequences with spectrum degeneracy at periodically located points

TL;DR

This paper provides new conditions under which sequences with spectrum degeneracy at periodically located points are predictable, enabling recovery of sequences from their periodic subsequences using linear predictors.

Contribution

It introduces sufficient conditions for predictability based on spectrum degeneracy at periodic points and demonstrates recoverability from subsequences for a dense class of sequences.

Findings

Sequences with spectrum degeneracy are predictable under certain conditions.

Periodic subsequences of these sequences are also predictable.

Sequences can be recovered from their periodic subsequences.

Abstract

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.