On nonlinear approximations of periodic functions of Besov classes using wavelet decomposi\-tions (Q1433483)

The paper deals with the so-called \(n\)-term approximation: given a normed linear space \(X\), a subset \(\Phi \subset X\), and an element \(f \in X\), one seeks a good approximation to \(f\) by a linear combination of \(\leq n\) elements \(\phi_k \in \Phi\) (the selected \(\{\phi_k\}\) may depend on \(f\)). \textit{D. Dung} [Vietnam J. Math. 26, No. 2, 165--176 (1998; Zbl 0921.46027)] and others introduced, for a given precompact subset \(W \subset X\), various non-linear \(n\)-widths associated with the \(n\)-term approximation. In the present paper, \(X=L_q({\mathbf T})\) and \(W\) is a Besov class \(SB_{p, \theta}^\omega\) defined with the help of a modulus of smoothness \(\omega\) satisfying certain conditions. The author proves that for \(1 \leq p,q \leq \infty\) all the \(n\)-widths considered decrease as \(\omega(1/n)\) when \(n \to \infty\). The upper estimate for the widths is obtained by means of a wavelet construction using the de la Vallée Poussin kernel. The case \(\omega(t)=t^\alpha\) was considered earlier by Dinh Dung.