Оцiнки апроксимацiйних характеристик i властивостi операторiв найкращого наближення класiв перiодичних функцiй у просторi

UDC 517.51 We obtain the exact-order estimates for orthoprojection widths and similar approximation characteristics of the Sobolev classes $W^{\boldsymbol{r}}_{p,\boldsymbol{\alpha}}$ and Nikol'skii–Besov classes $B^{\boldsymbol{r}}_{p,\theta}$ of periodic functions of one and several variables in the norm of the space $B_{1,1}$. In addition, we establish that the sequence of norms of linear operators that realize the orders of the best approximation of the classes $B^{\boldsymbol{r}}_{1,\theta}$ in space $B_{1,1}$ using trigonometric polynomials with ``numbers'' of harmonics from step hyperbolic crosses is unbounded in the multidimensional case.