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Exact asymptotic orders of various randomized widths on Besov classes

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This work is supported by National Nature Science Foundation of China [Grant Nos.11271199, 11671213]
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  • We study the efficiency of the approximation of the functions from the Besov space $ B_{p\theta}^\Omega(\mathbf{T}^d) $ in the norm of $ L_q(\mathbf{T}^d) $ by various random methods. We determine the exact asymptotic orders of Kolmogorov widths, linear widths, and Gel'fand widths of the unit ball of $ B_{p\theta}^\Omega(\mathbf{T}^d) $ in $ L_q(\mathbf{T}^d) $. 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 $ n^{-1/2} $ roughly.

    Mathematics Subject Classification: Primary: 41A46, 41A63; Secondary: 65C05, 65D99.

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