Pozharska K. Best approximations and entropy numbers of the classes of periodic multivariate functions

The thesis is devoted to the establishment of order estimates for the best approximations and entropy numbers of the classes of periodic multivariate functions. Namely, the best orthogonal and M-term trigonometric approximations are being studied for the periodic multivariate functions $D^{ \boldsymbol{ \psi } }_{{ \boldsymbol{ \beta } }}$, classes $L^{ \boldsymbol{ \psi } }_{{ \boldsymbol{ \beta } },1}$, and for the classes $L^{ \boldsymbol{ \psi } }_{{ \boldsymbol{ \beta } },p}$ of functions of small smoothness in the space $L_q$. Also, order estimates for the best bilinear approximations are received of the classes of periodic multivariate functions formed by shifts of an argument of functions from the classes $L_{{ \boldsymbol{\beta} },p }^{{ \boldsymbol{\psi} } }$, in the space $L_{q_1, q_2}$. Beside this, the entropy numbers are investigated of the classes $B^{ \Omega }_{ p, \theta }$ of periodic functions of one and many variables in the space $L_q$.