Filtration of linear functionals of periodically correlated sequences

The problem of the optimal estimation is considered for the linear functional \[ A{\zeta }=\sum _{j=0}^\infty {a}(j){\zeta }(-j) \] that depends on unknown values of a periodically correlated stochastic sequence $\zeta (j)$; the estimator is constructed from observations of the sequence $\zeta (j)+\theta (j)$, $j\leq 0$, where $\theta (j)$ is a periodically correlated noise. We obtain the mean square error and spectral characteristic of the optimal linear estimate of the functional $A{\zeta }$ in the case where the spectral densities of the sequences that generate $\zeta (j)$ and $\theta (j)$ are known. For the case where these spectral densities are unknown but a set of admissible spectral densities is given, we find the least favorable spectral density and minimax spectral characteristic for the optimal estimate of the functional $A{\zeta }$.