This article is cited in 9 papers

Convergence of Greedy Algorithms in Banach Spaces

E. D. Livshits

M. V. Lomonosov Moscow State University

Abstract: 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 $\ell^p$, $p\ge2$.

UDC: 517.518.843+519.651

Received: 10.12.2001
Revised: 11.04.2002

DOI: 10.4213/mzm195

 English version: Mathematical Notes, 2003, 73:3, 342–358

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