Minimax Estimation of Linear Functionals in the Presence of Two-Dimensional Noise

Problems of linear minimax estimation of linear functions from observations in a Gaussian random field are considered. The results of [I. A. Ibragimov and R. Z. Khas'minskii, Teor. Veroyatn. Primen., 29, No. 1, 19–32 (1984); 32, No. 1, 35–44 (1987)] are extended to this case. As an example, we examine minimax estimation of the value of the function $f(t,s)$ and its derivatives $\partial^{\alpha}f(t, s)/\partial t^{\alpha_1}\partial s^{\alpha_2}$. It is shown that the problems of estimation of a certain class of unbounded (in $L_2$) linear functions from observations in random fields with correlation operators $I$ and $R$ are equivalent in a certain sense if $R =I+K$, where $I$ is the identity operator and $K$ is a completely continuous operator.