Interpolation of stationary sequences observed with a noise

The problem of optimal linear estimation of the functional \[ A_s\xi =\sum _{l=0}^{s-1}\sum _{j=M_l}^{M_l+N_{l+1}}a(j)\xi (j), \qquad M_l=\sum _{k=0}^l (N_k+K_k), \qquad N_0=K_0=0, \] which depends on unknown values of a stochastic stationary sequence $\xi (j)$ with the help of observations of the sequence $\xi (j)+\eta (j)$ at points $j\in \mathbb {Z}\setminus S$, where $S=\bigcup _{l=0}^{s-1}\{ M_{l}, \dots , M_{l}+N_{l+1} \}$, is considered under the assumption that the sequences $\{\xi (j)\}$ and $\{\eta (j)\}$ are mutually uncorrelated. Formulas for calculating the mean-square error and spectral characteristic of the optimal linear estimator of the functional are proposed under the condition of spectral certainty, where both spectral densities of the sequences $\xi (j)$ and $\eta (j)$ are known. The minimax (robust) method of estimation is applied in the case where the spectral densities of the sequences $\xi (j)$ and $\eta (j)$ are not known exactly, but the sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some special sets of admissible densities.