On nonlinear approximation of multivariate functions with a mixed smoothness (Q1388214)

Let \(W\) be a compact subset in the normed linear space \(X\). The non-linear \(n\)-width is defined by \(\delta_n(W,X)=\inf_{F,M}\sup_{x\in W} \| x-M(F(x))\|\), where the infimum has taken over all continuous mappings \(F\) from \(W\) into \({\mathbb{R}}^n\) and all continuous mappings \(M\) from \({\mathbb{R}}^n\) into \(X\). For \(\alpha \in {\mathbb{R}}^d\) with \(\alpha_j > 0\), \(j=1,\dots ,n\), let \(BW_p^{\alpha}\) be the set of all functions \(f\) on the \(d\)-dimensional torus \({\mathbf T}^d\) such that the mixed fractional derivative in the sense of Weil \(f^{(\alpha)}\) satisfies the condition \(\| f^{(\alpha)}\|_{L_p({\mathbf T}^d)} \leq 1\), and let \(BH_p^{\alpha}\) be the set of all functions \(f\) such that the mixed higher-order difference \(\Lambda_h^r f\), \(h\in {\mathbf T}^d\) satisfies the condition \(\| \Lambda_h^r f\|_{L_p({\mathbf T}^d)} \leq \prod^d_{j=1} | h_j|^{\alpha_j}\) for some \(r\in {\mathbb{N}}^d\) with \(r_j>\alpha_j\). The author finds some estimates of the asymptotic degrees of the non-linear \(n\)-width \(\delta_n\) and the Aleksandrov \(n\)-width \(a_n\) of classes \(BW_p^{\alpha}\) and \(BH_p^{\alpha}\) in the space \(L_q({\mathbf T}^d)\), where \(1<p,q<\infty\). No proofs are given.