Convergence of a weak greedy algorithm when one vector is added to the orthogonal dictionary

Convergence of Weak Greedy Algorithms (WGA) and Weak Orthogonal Greedy Algorithms (WOGA) is studied for the subspace $\ell_1\subset\ell_2$ and dictionaries obtained from the standard orthogonal basis by additing one vector. It is shown that the condition on a weakening sequence sufficient for convergence of WOGA in the case of the orthogonal dictionary and an approximated element from $\ell_1$ is not applicable for these extensions of the dictionary. However, if a finite vector is added to the standard orthogonal dictionary, then the condition applicability holds. Similar results are presented for WGA. It is also shown that adding a vector even from $\ell_1$ to the standard orthogonal dictionary can significantly reduce the convergence rate of the Pure Greedy Algorithm (PGA).