Approximation characteristics of the Nikol'skii-Besov-type classes of periodic functions of several variables in the space $B_{q,1}$

Abstract

We obtained the exact order estimates of approximation of periodic functions of several variables from the Nikol'skii-Besov-type classes $B^{\Omega}_{p,\theta}$ by using their step hyperbolic Fourier sums in the space $B_{q,1}$. The norm in this space is stronger than the $L_q$-norm. In the considered situations, approximations by the mentioned Fourier sums realize the orders of the best approximations by polynomials with "numbers" of harmonics from the step hyperbolic cross. We also established the exact order estimates of the Kolmogorov, linear and trigonometric widths of classes $B^{\Omega}_{p,\theta}$ in the space $B_{q,1}$ for certain relations between the parameters $p$ and $q$.