On asymptotic orders of \(n\)-term approximations and non-linear \(n\)-widths
Let \(\Phi= \{ \varphi_{k}\}_{k=1}^{\infty}\) be a family of elements of a normed linear space \(X\). Denote by \(M_{n}(\Phi)\) the nonlinear manifold of all linear combinations of \(n\) elements of \(\Phi\), \(\varphi =\sum_{k=1}^{n} a_k\varphi_k\). For a given subset \(W\) in \(X\) the best \(n\)-term approximation \(\sigma_n(W, \Phi, X)\) by the family \(\Phi\) is given by \(\sigma_n(W, \Phi, X)=\sup_{f\in W}\inf_{\varphi \in M_n(\Phi)} \|f - \varphi \|\). Let denote \({\mathcal{ F}}(X)\) the set of all bounded \(\Phi\) whose intersection \(\Phi\cap L\) with any finite-dimensional subspace \(L\) in \(X\) is a finite set. The nonlinear \(n\)-width \(\tau_n(W,X)\) is defined by \(\tau_n(W,X)=\inf_{S,\Phi}\sup_{f\in W}\|f-S(f)\|\), where the infimum is taken over all continuous mappings \(S\) from \(W\) into \(M_n(\Phi)\) and all families \(\Phi\in{\mathcal{F}}(X)\). Similar to \(\tau_n(W,X)\) is the nonlinear \(n\)-width \(\tau'_n(W,X)\) which is defined in the same way as \(\tau_n(W,X)\) but the infimum is taken over all continuous mappings \(S\) from \(W\) into a finite subset of \(M_n(\Phi)\). Let \(M_n\) be the subset in \(l_{\infty}\) of all sequences \(x\in l_{\infty}\) for which \(x_k=0\), \(k\in Q\), for some set of natural numbers \(Q\) with cardinality \(|Q|=n\). The nonlinear \(n\)-width \(\alpha_n(W,X)\) is given by \(\alpha_n(W,X)=\inf_{G,\Phi}\sup_{f\in W} \|f-R_{\Phi}(G(f))\|\), where \(R_{\Phi}\) is the mapping from \(M_n\) into \(X\) defined by \(R_{\Phi}(x)=\sum_{k\in Q}x_k\varphi_k\), \(\varphi_k\in\Phi\), the infimum is taken over all continuous mappings \(G\) from \(W\) into \(M_n\) and all bounded families \(\Phi\) in \(X\). In this paper the author gives the asymptotic orders of the best \(n\)-term approximation \(\sigma_n(W,\Phi,X)\) and nonlinear widths \(\tau_n(W,X)\), \(\tau'_n(W,X)\) in the case, where \(X\) is the \(L_q(T^d)\) on the \(d\)-dimensional torus \(T^d\); the set \(\Phi\) consists of the mixed dyadic scales of the tensor product multivariate de la Vallée Poussin kernel; and the unit ball \(SB^r_{p,\theta}\) of the Besov space with common mixed smoothness \(r\) is taken as the \(W\). These results were earlier proved in [J. Approximation Theory 102, No. 2, 217-242 (2000; Zbl 0951.41011)] under stronger restrictions on \(p\), \(r\) and \(\theta\). No proofs are given.
scientific article; zbMATH DE number 1485037