Estimation of periodically correlated random fields that are isotropic on a sphere

The problem of optimal linear estimation of functionals depending on the unknown values of a spatial temporal isotropic random field $\zeta(j,x)$, which is periodically correlated
with respect to discrete time argument $j\in\mathrm Z$ and mean-square continuous isotropic on the unit sphere ${S_n}$ with respect to spatial argument $x\in{S_n}$. Estimates are based on observations of the field $\zeta(j,x)+\theta(j,x)$ at points $(j,x):$ $j\in Z\backslash\{0, 1, .... , N\}$, $x\in S_{n}$,
where $\theta(j,x)$ is an uncorrelated with $\zeta(t,x)$ spatial temporal isotropic random field, which is periodically correlated
with respect to discrete time argument $j\in\mathrm Z$ and mean-square continuous isotropic on the sphere ${S_n}$ with respect to spatial argument $x\in{S_n}$. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of the functional are derived in the case where the spectral density matrices are exactly known. Formulas that determine the least favourable spectral density matrices and the minimax (robust) spectral characteristics are proposed in the case where the spectral density matrices are not exactly known but a class of admissible spectral density matrices is given.