Some remarks on an interpolation problem of A. M. Yaglom

The paper deals with the problem of best mean-square interpolation of a continuous $q$-variate weakly stationary random process $\textbf x$ over $\mathbf R$ on the basis of the values $x_k$, $k\in\mathbf Z$, which was studied first by A. M. Yaglom [Uspehi Matem. Nauk (N.S.), 4 (1949), pp. 173–178] in the case $q=1$. For the family $\mathscr J_\mathbf{Z}$ of all subsets of $\mathbf R$ which are shifts of $\mathbf Z$, criterions of $\mathscr J_\mathbf Z$-singularity and $\mathscr J_\mathbf Z$-regularity in terms of the nonstochastic spectral measure of $\mathbf x$ are given. Related results for stationary sequences over $\mathbf Z$ are stated.