On \(n\)-term approximation with positive coefficients
The paper considers the Pure Greedy Algorithm (PGA) and the Weak Greedy Algorithm (WGA), and presents algorithms for constructing \(n\)-terms approximations with nonnegative coefficients. The convergence theorem is proved for a~``positive'' analog of the PGA. A condition on the sequence of weakness coefficients that is sufficient for the convergence of the Positive WGA is established as follows: Suppose that we are given a weakness sequence \(\tau= \{T_m\}^\infty_{m=1} \subset [0,1]\}\) such that \(\sum t_m/m <\infty\). Then there is a~positive dictionary \({\mathcal D}\) of a~real separable Hilbert space~\(H\) and a~target function \(f_0\in H\) such that there exists an~implementation of the PWGA \(\{G^{\text{WGA}} _m +(f_0,{\mathcal D},\tau)\}^\infty_{m=0}\) that fails to converge to~\(f_0\) as \(m\to \infty\). This condition is also necessary for the class of monotone sequences.
scientific article; zbMATH DE number 5319843