Error bounds of Approximate Weak Rescaled Pure Greedy Algorithms

We study the greedy algorithms for [Formula: see text]-term approximation. We propose a modification of the Weak Rescaled Pure Greedy Algorithm (WRPGA) — Approximate Weak Rescaled Pure Greedy Algorithm (AWRPGA) — with respect to a dictionary of a Banach space [Formula: see text]. By using a geometric property of the unit sphere of [Formula: see text], we obtain a general error estimate in terms of some [Formula: see text]-functional. This estimate implies the convergence condition and convergence rate of the AWRPGA. Furthermore, we obtain the corresponding error estimate for the Vector Approximate Weak Rescaled Pure Greedy Algorithm (VAWRPGA). We show that the AWRPGA (VAWRPGA) performs as well as the WRPGA (VWRPGA) when the noise amplitude changes relatively little. Finally, by using the wavelet bases and trigonometric system of Lebesgue spaces, we show that the convergence rate of the AWRPGA is optimal.