Voitovych K. Approximation and asymptotic properties of functions from Hardy spaces on some domains

The thesis is devoted to questions of approximation and asymptotic properties of functions in Hardy spaces and Paley - Wiener spaces. We conducted a review of the literature and describes the important facts from classic Hardy spaces, weighted Hardy spaces, Paley - Wiener spaces, signal processing, Wiener filtering theory and Hilbert transform. In second section we obtained the solvability criterion for the decomposition problem for functions in the Paley - Wiener space. This problem is in finding of two functions, each of them being ”large” only in upper and lower half-planes and sume of above functions is an initial function. Consequently, it is established that there exists the solutions of the decomposition problem if all Fourier coefficients with positive numbers are equal to zero; we obtained a criterion of the solution of the decomposition problem in the angle for function under the condition of a certain regularity of coefficients; obtained the solution of the decomposition problem for functions with small exponential type in half - plane. The third section deals with Wiener filtering theory. We consider the analogue of the classic Wiener filtering theory to a half-strip of complex domain. We analyzed the major problem of signal processing: to determine an unknown filter (”black box”); in particular, to reconstruct, if possible, a filter knowing the energy densities of an input-output pair. We proved that if signal does not admit a holomorphic continuation as an entire function or it admit a holomorphic continuation, but this continuation is extremely large, then there exists the solution of the filtering identification problem. In fourth section is obtained a boundedness criterion for the Hilbert transform on the Paley – Wiener space. Two simple methods of evaluation of the Hilbert transform were shown in the research basing on the reserved result.