Shchestuk N. Estimation of functionals of homogeneous random firlds under uncertainties

The dissertation is devoted to the interpolation, extrapolation and filtration problems of functionals of homogeneous random fields under uncertainties. It is developed method of Kolmogorov for the problem of optimal linear estimation of the unknown values of homogeneous random fields from observations of the field with additive noise. This method is based on the properties of Hilbert space projections and Fourier transformations. The spectral characteristics and the mean square error of the optimal estimate of functionals on the unknown values of homogeneous random fields may be calculated by this method under conditions that spectral densities are known exactly. In the case where spectral densities are not known exactly, but sets of possible spectral densities are given, the minimax (robust) approach to estimate unknown values of the fields is applied. With the help of this approach the least favorable spectral densities and the minimax spectral characteristic of the optimal linear estimates for someclasses of random fields are found. Algorithm and computer realization for interpolation problem is represented. Estimation problems are investigated for homogeneous random fields of discrete argument, mean square continuous fields and for continuous fields from observations at discrete points.