Comparison of the Convergence Rate of Pure Greedy and Orthogonal Greedy Algorithms

The following two types of greedy algorithms are considered: the pure greedy algorithm (PGA) and the orthogonal greedy algorithm (OGA). From the standpoint of estimating the rate of convergence on the entire class $\mathscr A_1(\mathscr D)$, the orthogonal greedy algorithm is optimal and significantly exceeds the pure greedy algorithm. The main result in the present paper is the assertion that the situation can also be opposite for separate elements of the class $\mathscr A_1(\mathscr D)$ (and even of the class $\mathscr A_0(\mathscr D)$): the rate of convergence of the orthogonal greedy algorithm can be significantly lower than the rate of convergence of the pure greedy algorithm.