Estimates of the characteristics of nonlinear approximation of classes $B^{\Omega}_{p,\theta}$ of periodic functions of many variables in the space $B_{q,1}$

Abstract

Exact order estimates of the best $m$-term trigonometric approximation and the best orthogonal trigonometric approximation of functions from the Nikol'skii-Besov-type classes $B^{\Omega}_{p,\theta}$ in the Lebesgue subspaces $B_{q,1}$ for certain relations between the parameters $p$ and $q$ are obtained. It is shown that in the considered cases the mentioned approximation characteristics of the classes $B^{\Omega}_{p,\theta}$ in the spaces $B_{q,1}$ and $L_q$ differ in order. In addition, it was found that for $1<q<p<\infty$, in contrast to the case $2<p<q<\infty$, the obtained orders of these quantities are realized by approximation of functions from the classes $B^{\Omega}_{p,\theta}$ by their step hyperbolic Fourier sums with a corresponding number of harmonics.