Exact asymptotic orders of various randomized widths on Besov classes

Abstract

We study the efficiency of the approximation of the functions from the Besov space \begin{document}$ B_{p\theta}^\Omega(\mathbf{T}^d) $\end{document} in the norm of \begin{document}$ L_q(\mathbf{T}^d) $\end{document} by various random methods. We determine the exact asymptotic orders of Kolmogorov widths, linear widths, and Gel'fand widths of the unit ball of \begin{document}$ B_{p\theta}^\Omega(\mathbf{T}^d) $\end{document} in \begin{document}$ L_q(\mathbf{T}^d) $\end{document} . Our results show that the convergence rates of the randomized linear and Gel'fand methods are faster than the deterministic counterparts in some cases. The maximal improvement can reach a factor \begin{document}$ n^{-1/2} $\end{document} roughly.