Minimax prediction problem for multidimensional stationary stochastic sequences

The considered problem is estimation of the unknown value of the functionate $A\vec{\xi}\sum^\infty_{j=0}\vec{a}(j)\vec{\xi}(j)$ and $A_N\vec{\xi}\sum^N_{j=0}\vec{a}(j)\vec{\xi}(j)$ which depend on the unknown values of a multidimensional stationary stochastic sequence $\vec{\xi}(j)$ based on observations of the sequence $\vec{\xi}(j), j<0,$ from the class $\Xi$ of sequences which satisfy conditions $E\vec{\xi}(j)=0, \|\vec{\xi}(j)\|^2\leq P.$ The maximum values of the mean-square errors of the optimal estimates of the functionals $A\vec{\xi}$ and $A_N\vec{\xi}$ are found. It is shown that these maximum values of the errors in the class $\Xi$ give the moving average sequences which are determined by eigenvectors of compact operators constructed with the help of the sequence $\vec{a}(j).$