Continuous algorithms in \(n\)-term approximation and nonlinear widths (Q1971918)

Let \(X \) be a quasi-normed linear space and \(\Phi = \{\phi_k\}_{k=1}^{\infty} \) a family of elements in \(X\). If the family \(\Phi \) is bounded, i.e., \(\|\phi_k\|\leq C\), \(k=1,2,\dots\), and the span of \(\Phi \) is dense in \(X\), then \(\Phi \) is called a dictionary. Denote by \(M_n(\Phi)\) the set of all linear combinations \(\phi =\sum_{k\in Q}a_k\phi _k\), where \(Q\) is a set of natural numbers with \(|Q|=n\). If \(W\) is a subset \(X\) then the quantity \(\sigma_n(W,\Phi ,X) = \sup_{f\in W}\inf_{\phi \in M_n(\Phi)} \|f-\phi \|\) is called the \(n\)-term approximation of \(W\) by the family \(\Phi \). In this paper the author considers optimal continuous algorithms in \(n\)-term approximation based on various non-linear \(n\)-widths, and the \(n\)-term approximation \(\sigma_n(W,V,X)\) by the dictionary \(V\) formed from the integer translates of the mixed dyadic scales of the tensor product multivariate de la Vallée Poussin kernel, for the unit ball of Sobolev and Besov spaces of functions with a common mixed smoothness. The asymptotic orders of these quantities are given. These orders are achieved by a continuous algorithm of \(n\)-term approximation by \(V\), which is explicitly constructed.