Approximation Properties of the Vector Weak Rescaled Pure Greedy Algorithm

Abstract

We first study the error performances of the Vector Weak Rescaled Pure Greedy Algorithm for simultaneous approximation with respect to a dictionary $\mathcal{D}$ in a Hilbert space. We show that the convergence rate of the Vector Weak Rescaled Pure Greedy Algorithm on ${\mathcal{A}}_1\left(\mathcal{D}\right)$ and the closure of the convex hull of the dictionary $\mathcal{D}$ is optimal. The Vector Weak Rescaled Pure Greedy Algorithm has some advantages. It has a weaker convergence condition and a better convergence rate than the Vector Weak Pure Greedy Algorithm and is simpler than the Vector Weak Orthogonal Greedy Algorithm. Then, we design a Vector Weak Rescaled Pure Greedy Algorithm in a uniformly smooth Banach space setting. We obtain the convergence properties and error bound of the Vector Weak Rescaled Pure Greedy Algorithm in this case. The results show that the convergence rate of the VWRPGA on ${\mathcal{A}}_1\left(\mathcal{D}\right)$ is sharp. Similarly, the Vector Weak Rescaled Pure Greedy Algorithm is simpler than the Vector Weak Chebyshev Greedy Algorithm and the Vector Weak Relaxed Greedy Algorithm.