Approximation properties of some vector weak biorthogonal greedy algorithms

We study the efficiency of the vector greedy algorithms for simultaneous sparse approximation in a Banach space. We employ a unified way of analyzing the approximation properties of the Vector Weak Chebyshev Greedy Algorithm, the Vector Weak Greedy Algorithm with Free Relaxation and the Vector Rescaled Weak Relaxed Greedy Algorithm. We obtain the necessary and sufficient conditions for the convergence and the optimal convergence rate on the basic sparse class for these algorithms. It shows that these vector greedy algorithms are simple but highly efficient in dealing with multi-target element.