Optimal Filtering of Square-Integrable Signals in Gaussian Noise

Expressions are considered for the optimal mean-square error $\delta^2$ of restoration of signals from $\mathrm L_2(0,T)$, in Gaussian noise $\xi_1(t)$ the exact asymptotic form of $\delta^2$ is obtained for the case in which $\xi_1(t)=\varepsilon\xi(t)$, $\varepsilon^2\to 0$, while $\theta(t)\in\mathfrak{A}$, $\mathfrak{A}$ is an ellipsoid in $\mathrm L_2(0,T)$, and also when $T\to\infty$. It is shown that the linear estimates are asymptotically optimal.