Extrapolation of periodically correlated stochastic processes observed with noise

We consider the problem of the optimal linear estimation of the functional \[ A\zeta =\int _0^\infty a(t)\zeta (t) dt \] depending on unknown values of a periodically correlated stochastic process $\zeta (t)$. An estimator is constructed from observations of the process $\zeta (t)+\theta (t)$ for $t<0$, where $\theta (t)$ is a periodically correlated process being uncorrelated with $\zeta (t)$. Formulas for calculating the spectral characteristic and the mean square error of the optimal linear estimator of the functional are proposed in the case where spectral densities are known. In the case where spectral densities are not known but a set of admissible spectral densities is specified, formulas that determine the least favorable spectral density and the minimax (robust) spectral characteristics of optimal estimators of the above functional are proposed.