INTERPOLATION PROBLEMS FOR RANDOM FIELDS FROM OBSERVATIONS IN AREAS THAT REPRESENT A SYSTEM OF EMBEDDED RECTANGLES

Forecasting of static processes and estimation of random fields of a different nature is becoming more widespread among scientists of different specialties, and a new branch of science appears with its specific methodology. That problems of estimation of the unknown values of random fields are generalization of problems of extrapolation, interpolation and filtering of stochastic processes. The study of the dependence of the obtained formulas on the geometry and the number of embeds are the topical problems in the field of the forecasting theory, in geology, geodesy, and some other directions. The methods of solution of linear estimation problems for stochastic processes and random fields were developed by A. M. Kolmogorov, A. M. Yaglom. Traditional methods of solution of these problems are employed under the condition that spectral densities are known exactly. The case of estimating the unknown values of a random field for an area that represents a system of embedded rectangles is of interest in the study of random fields with peculiarities. The problem is the estimation of linear functionals which depend on the unknown values of a homogeneous random field ξ(x, y) in the region K from observations of ξ(x, y) at points (x, y) ∈ Z 2 \ K, where K is a region that represents the union of the edges of the rectangles mx × my, with the number of rectangles — sx, lx and ly spaced between the attachments on the X and Y axes, respectively. That is, we find a value A？ kξ from the class of linear functionals, which minimizes the value of the mean square error ？ = M|Akξ ？ A？ kξ|2. To solve this problem, we used a classical method of projections in the Hilbert space. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case when the areas of observations represent a system of embedded rectangles.