Optimal Nonlinear Extrapolation, Filtering, and Interpolation of Functions of Gaussian Processes

An optimal (in the sense of mean-square deviation) prognosis of the values of a function of a stationary Gaussian process $f(x_{t+\tau})$, $\tau>0$, is constructed from known values of the process $x_s$, $s\leq t$. The more general problem of optimal prognosis of $f(x_{t+\tau})$ from known values of a process $z_s$, $s\leq t$, stationarily related to $x_t$ is solved. The conditions are analyzed for the error-free interpolation of an unknown value $f(x_t)$, $t\in U$, from values of the process $x_s$ known on the entire number line excluding the interval $U$.