Kolmogorov widths between the anisotropic space and the space of functions with mixed smoothness (Q1433346)

Let \(W^R_p\), \(1\leq p\leq\infty\), \(R\in{\mathbb R}^d\), \(R>0\), be the anisotropic Sobolev space of functions on a \(d\)-dimensional torus \([0, 2\pi]^d\), let \(W^r_{\text{mix}, p}\), \(r\in {\mathbb R}^d\), \(r>0\), be the Sobolev space of functions with mixed derivatives, let \(H^R_p\) be the anisotropic Hölder-Nikolskii space, and let \(H^r_{\text{mix}, p}\) be the Hölder-Nikolskii space of functions with mixed difference, understood in usual sense [see \textit{V. N. Temlyakov}, Approximation of periodic functions (1993; Zbl 0899.41001)]. Let \(F^r_{\text{mix}, p}\) denote one of the spaces with mixed smoothness \(W^r_{\text{mix}, p}\), \(H^r_{\text{mix}, p}\), let \(F^R_p\) denote one of the anisotropic spaces \(W^R_p\), \(H^R_p\) and let \(BF^r_{\text{mix}, p}\), \(BF^R_p\) be the unit balls of the spaces \(F^r_{\text{mix}, p}\), \(F^R_p\), respectively. The authors consider the Kolmogorov \(M\)-width of the classes \(BF^r_{\text{mix}, p}\) in the space \(F^R_p\) and that of the classes \(BF^R_p\) in the space \(F^r_{\text{mix}, p}\). They find the asymptotic order of the widths and give weakly asymptotic optimal approximation subspaces which realize the order of widths. The following main results are given: Theorem 1. Let \(1\leq p\leq\infty\), \(r-R>0\), \({\overline \alpha}:=\min_{1\leq i\leq d}\alpha_i\), \(\alpha_i=r_i-R_i\), \(i=1,\dots, d\). Then \(d_M(BF^r_{\text{mix}, p}, F^R_p)\asymp M^{-{\overline \alpha}}\). Theorem 2. Let \(1\leq p\leq\infty\), \(v:=\sum^d_{i=1}r_i/R_i <1\), \(g(R):=(\sum^d_{j=1} R^{-1}_j)^{-1}\). Then \(d_M( BF^R_p, F^r_{\text{mix}, p})\asymp M^{-(1-v)g(R)}\).