Investigation of a New Class of Integral Equations and Applications to Estimation Problems (Filtering, Prediction, System Identification)

Kolmogorov [1] initiated the study of filtering and extrapolation of stationary time series. These and other related problems were studied by N. Wiener in 1942 for stationary random processes and his results were published later in Wiener [1]. The basic integral equation of the theory of stochastic optimization for random processes is (1.1)$$ Rh = \int_{{t - T}}^t {R(x,y)h(y)dy = f(x),\quad t - T \leqslant x \leqslant t} $$ where R(x,y) is a nonegative definite kernel, a correlation function, f(x) is a given function, and T > 0 is a given number. In Wiener [1] equation (1.1) was studied under the assumptions that R(x,y) = R(x-y) and T = +∞. We note that in applications T is the time of signal processing and the assumption about the kernel means that only stationary random processes were studied in Wiener [1]. Under these and some additional assumptions concerning the kernel R(x) a theory of the integral equation (1.1), now widely known as the Wiener-Hopf method, was given in Wiener-Hopf [1]. Their results were developed later in Krein [1], Gohberg-Krein [1], and Gohberg-Feldman [1].