The δ-Simultaneous Super Algorithms in Hilbert Spaces

We propose the [Formula: see text]-Simultaneous Orthogonal Super Greedy Algorithm ([Formula: see text]-SOSGA), the [Formula: see text]-Simultaneous Super Greedy Algorithm with Free Relaxation ([Formula: see text]-SSGAFR) and the [Formula: see text]-Simultaneous Rescaled Pure Super Greedy Algorithm ([Formula: see text]-SRPSGA) for a certain finite number of target functions with respect to a dictionary [Formula: see text] in a Hilbert space. We estimate the upper bounds of the error for these algorithms when [Formula: see text] is a Riesz dictionary. The results show that the convergence rates of the [Formula: see text]-SOSGA, the [Formula: see text]-SSGAFR and the [Formula: see text]-SRPSGA on the closure of the convex hull of [Formula: see text] are optimal. Based on the joint greedy selection step, the [Formula: see text]-simultaneous super algorithms exhibit greater adaptivity than the super greedy algorithms. Our numerical experiments verify the efficiency and superiority of these algorithms over some existing well-known super greedy algorithms.