Gunko M. Problems of optimal recovery of polylinear functionals and operators on linear information

The dissertation work reflects the investigation of optimal restoration of polylinear functionals and operators on linear information. The optimal linear information, optimal methods of recovery are found, and the optimal errors of recovery of scalar products, bilinear, n-linear functionals for concrete functional classes, and classes of elements of separable Hilbert space over a field of real or complex numbers are calculated. The optimal linear information, optimal recovery methods are found, and the optimal errors of recovery of n-function convolutions for specific functional classes are calculated. The problem of optimal recovery of subsets of Hilbert space, which are the image of a sphere of the unit radius for the action of a compact operator, is solved. It is done based on information about the values of the first few Fourier coefficients that are set not accurately according to some operator-related orthonormal system. The problem of optimal recovery of a scalar product on a Cartesian product of subsets of Hilbert space is solved. One of which is the image of a sphere of the unit radius for the action of a compact operator, and the other one is with the image of a sphere of the unit radius for the action of a bounded operator of special structure based on the information with an error about the first few Fourier coefficients of the elements of these subsets.