Greedy algorithms with prescribed coefficients (Q996000)

A symmetric subset \(D\) of the unit ball of a Banach space \(X\) is called a dictionary if its linear span is dense in \(X\). The primary goal of the article under review is to study representations of an element \(f \in X\) by a series, \[ f \sim \sum_{j=1}^\infty c_j(f)g_j (f), \quad g_j(f)\in D, \quad c_j(f)>0, \quad j=1, 2,\dots. \] The sequences \(c_j(f)\) and \(g_j(f)\) are constructed inductively by some greedy algorithm. A surprising fact, discovered by the author, is that under certain conditions one can choose the coefficients \(c_j\) in advance, independently of \(f\). For example, let \(X\) be a uniformly smooth Banach space with modulus of smoothnes \(\rho(u)\) and let \(c_k\) be any sequece of positive numbers for which \(\sum_{k=1}^\infty c_k=\infty\) and \(\sum_{k=1}^\infty \rho(\theta c_k)<\infty\) for any \(\theta >0\). Set \(f_0=f\) and for \(m \geq 1\) find \(\phi_m \in D\) and \(f_m\) inductively from the conditions \[ \| f_{m-1}-c_m\phi_m\| =\inf_{g \in D}\| f_{m-1}-c_m g\| , \] \(f_m=f_{m-1}-c_m\phi_m\). Then \(\lim \inf \| f_m\| =0\) for any \(f\) when \(m \to \infty\). The above described algorithm is called \(X\)-greedy. A number of other greedy algorithms are considered and their convergence and rate of convergence are studied.