Entropy meaning of summability of the logarithm

We consider the mutual relations between the concepts of sets of uniqueness for analytic functions, the loss of entropy in nondetermined stationary linear filters, Szegö's theorem and the familiar condition of summability of the logarithm. The goal of the paper is to give the physical meaning of these mutual relations. Here we take the concept of linear stationary filter and loss of entropy in it as basic. In the first part of the paper the account is given for the case of discrete time, and in the second part we give the method of passing to continuous time. To this end we introduce the concept of stationary sampling system. This is a sequence of functions from $L^2(\mathbf R)$, which transforms any stationary Gaussian process $(\mathfrak X_t)_{t\in\mathbf R}$ with continuous correlation function into a stationary Gaussian process with discrete time $Y_n\overset{\operatorname{def}}=\int_{\mathbf R}\varphi_n\mathfrak X_tdt$, $n\in\mathbf Z$. Such systems can be described in terms of the Fourier transform. Laguerre systems $\varphi_n(x)=\sqrt{\dfrac{\operatorname{Im}z}{\pi}}\cdot\dfrac{1}{x-z}\biggl(\dfrac{x-z}{x-z}\biggr)^n$ where $z$ is a fixed point in the upper half-plane, play a special role among all sampling systems. If $z=i$, then $\varphi_n$ is a classical Laguerre function on the line up to a multiplicative constant. Laguerre sampling systems allow one to give the entropy meaning to the values of the harmonic extension to the upper half-plane of the logarithm of the spectral density of the process $(\mathfrak X_t)_{t\in\mathbf R}$.