Treffer: Simultaneous greedy approximation in Banach spaces

The notion of the weak greedy algorithms (WGA) in the case of Hilbert space \(H\) was introduced and studied by the second author [Adv. Comp. Math. 12, 213--227 (2000; Zbl 0964.65009)]. This algorithm is provided for each \(f\in H\) and any dictionary \(\mathcal{D} \) by the sums \(G_m(f,\mathcal{D})\) \(=\sum _{j=1}^mc_j\varphi _j(f),\) where \(m\in N \) and \(c_j\) are some numbers which depend on \(f\) and \(\varphi _j(f)\in \mathcal{D} \). Later it has been introduced the weak orthogonal greedy algorithm (WOGA) in order to enhance the rate of convergence of greedy algorithms. Recently, the authors studied analogues of WGA for a given finite number of functions \(f^1,\dots ,f^N \) with a requirement that the dictionary elements \(\varphi _j\) are the same for all \(f^i,\;i=1,\dots ,N.\) They have studied convergence and rate of convergence of the introduced algorithms which are called simultaneous algorithm. The goal of the present paper is twofold. First the authors work in a Hilbert space and enhance the convergence of the simultaneous greedy algorithms by introducing an analogue of orthogonal process. Then they study simultaneous greedy approximation in a more general setting, namely, in uniformly smooth Banach spaces.