Shidlich A. Extremal Problems of Approximation Theory in Functional Spaces

The thesis is devoted to the investigation of important extremal problems of approximation theory in functional spaces.The main results of the thesis are as follows: We found the exact values and the exact order estimates of best approximations of integrals by integrals of finite rank of some functional classes. There are found the exact order estimates of important nonlinear approximative characteristics of classes of functions of several variables in the integral metric and in the spaces S^p. We obtain the direct and inverse approximation theorems of functions of one and several variables by Taylor-Abel-Poisson operators. We calculated the exact values of the best n-term approximations of the sets of the images of the diagonal operators in the sequences Orlicz spaces. New Chebyshev type inequalities are proved. There are established new criteria for periodic functions real-valued on the real axis to belong to the set of infinitely differentiable functions, the set of functions analytic on the axis and to the set of entire functions.