Chaichenko S. Extremal Problems of Theory of Approximation of Functions of Real and Complex Variables

The thesis is solved a number of important extremal problems of the theory of approximation of functions defined on the real axis (integrated periodic or locally Lebesgue integrated) and functions defined in the domains of the complex plane - the upper half-plane and the unit disc. The main results of the thesis are as follows: 1. Sufficient conditions of convergence in integral metrics and the pointwise convergence of Fourier series on the Takenaka-Malmquist systems on the real axis are found. 2. We calculated the exact value of the best approximation of the Cauchy kernel in the integral metric by certain subspaces and we constructed in the explicit form for these spaces the element of the best approximation and proved the uniqueness of its. 3. It is also constructed the best linear method of pointwise and uniform approximation of the functions in the unit ball of the Hardy spaces. It is shown that this method is the best interpolation method of the approximation. 4. There are found the asymptotic equalities for the exact upper bounds of the deviations of Vallee Poussin sums and some special trigonometric polynomials on the classes periodic functions. All received equality provide the solution of the Kolmogorov-Nikol'skii problem. Also, it is calculated the value of the least upper bound pointwise deviation of Vallee Poussin sums on the class of analytic in the unit disk functions that are represented by the Cauchy integrals. 5. We describe all cases of the relations between the parameters of the sieve-operator F. Wiener, when the norm of the operator is equivalent to 1. We have used these results to find the best approximations of the functions of Hardy spaces. 6. We have proved some direct and inverse theorems of the approximation theory in Lebesgue spaces with variable exponent and Orlicz spaces. 7. There are calculated the exact values of the best approximations, the base widths and the Kolmogorov widths of the sets of the images of the diagonal operators in the sequences Orlicz spaces.