Let \(X\) be a real Banach space. Let \({\mathcal D} \subset X\) be a given dictionary, i.e., \(\|g\| \le 1\) for each \(g \in \mathcal D\) and the closure of \(\mathrm{span}\,\mathcal D\) is equal to \(X\). Then let \(F\) be the set \[ \{f \in X:\, f = \sum_{j=1}^{\infty} c_j\,g_j\,, \;g_j\in \mathcal D\,,\;\sum_{j=1}^{\infty} |c_j| \le 1\} \] or its closure.
In this paper, the asymptotic behavior of sparse approximation of elements \(f\in F\) is studied. The authors discuss the classical question for sparse approximation: Is it true that for any \(f\in F\) the best \(m\)-term approximation with respect to \(\mathcal D\) decays faster than the corresponding supremum over \(F\)? The results are formulated in terms of modulus of smoothness of \(X\). It is shown that for any \(f \in F\) one can improve the upper bound on the rate of convergence of the approximation error by a greedy algorithm, if some information from the previous iterations of the algorithm is used.