Shkapa V. Best approximations and greedy algorithms of classes of (psi,beta)-differentiable functions

The thesis investigates some approximate characteristics of the classes L_{beta, p}^psi of periodic functions in the Lebesgue spaces. It sets exact order estimates of the best approximations, approximations by Fourier sums and the best orthogonal trigonometric approximations of 2pi periodic functions F_{psi}(x,beta) in the space L_q by 1<q<infty. The thesis finds exact order estimates of the best approximations, approximations by Fourier sums, the best m-term trigonometric approximations, the best orthogonal trigonometric approximations, greedy algorithms and trigonometric widths of the classes L_{beta, p}^psi in the space L_q for some relations between parameters p and q. It obtains exact order estimates of the best bilinear approximations of the classes of functions of two variables generated by functions of a single variable from the classes L_{beta, p}^psi by the shifts of the argument in the space L_{q_1,q_2}, 1 leq q_1, q_2 leq infty. The fact that the order estimates of the best bilinear approximations coincide with the order estimates of the Kolmogorov widths that were obtained before was revealed during the research.