Geodesic
Convergence of Greedy Algorithms in Banach Spaces
Matematičeskie zametki, Tome 73 (2003) no. 3, pp. 371-389

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We study the convergence of greedy algorithms in Banach spaces. We construct an example of a smooth Banach space, where the $X$-greedy algorithm converges not for all dictionaries and initial vectors. We also study the $R$-greedy algorithm, which, along with the $X$-greedy algorithm, is a generalization of the simple greedy algorithm in Hilbert space. We prove its convergence for a certain class of Banach spaces. In particular, this class contains, the spaces $\ell^p$, $p\ge2$. 
@article{MZM_2003_73_3_a4,
     author = {E. D. Livshits},
     title = {Convergence of {Greedy} {Algorithms} in {Banach} {Spaces}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {371--389},
     publisher = {mathdoc},
     volume = {73},
     number = {3},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2003_73_3_a4/}
}
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E. D. Livshits. Convergence of Greedy Algorithms in Banach Spaces. Matematičeskie zametki, Tome 73 (2003) no. 3, pp. 371-389. http://geodesic.mathdoc.fr/item/MZM_2003_73_3_a4/