On the Szegő—Kolmogorov prediction theorem

The classical Szegő—Kolmogorov theorem characterizes the weights ω such that the family of exponentials with positive integer frequencies spans the whole weighted space L²(T, w) on the circle. Kolmogorov’s probabilistic interpretation of this result connects it with the possibility to ‘predict precisely the future from the past’ for the stationary stochastic processes with discrete time. We discuss the problem whether the prediction remains possible if some part of the ‘past’ is not known.