Geodesic
Convergence of a weak greedy algorithm when one vector is added to the orthogonal dictionary
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2022), pp. 17-25

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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). 
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     author = {A. S. Orlova},
     title = {Convergence of a weak greedy algorithm when one vector is added to the orthogonal dictionary},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {17--25},
     publisher = {mathdoc},
     number = {5},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2022_5_a2/}
}
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A. S. Orlova. Convergence of a weak greedy algorithm when one vector is added to the orthogonal dictionary. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2022), pp. 17-25. http://geodesic.mathdoc.fr/item/VMUMM_2022_5_a2/