Weak greedy algorithms and the equivalence between semi-greedy and almost greedy Markushevich bases
Abstract
We introduce and study the notion of weak semi-greedy systems -which is inspired in the concepts of semi-greedy and Branch semi-greedy systems and weak thresholding sets-, and prove that in the context Markushevich bases in infinite dimensional Banach spaces, the notions of \textit{ semi-greedy, branch semi-greedy, weak semi-greedy, and almost greedy} Markushevich bases are all equivalent. This completes and extends some results from \cite{Berna2019}, \cite{Dilworth2003b}, and \cite{Dilworth2012}. We also exhibit an example of a semi-greedy system that is neither almost greedy nor a Markushevich basis, showing that the Markushevich condition cannot be dropped from the equivalence result. In some cases, we obtain improved upper bounds for the corresponding constants of the systems.
Keywords
Cite
@article{arxiv.2004.06849,
title = {Weak greedy algorithms and the equivalence between semi-greedy and almost greedy Markushevich bases},
author = {Miguel Berasategui and Silvia Lassalle},
journal= {arXiv preprint arXiv:2004.06849},
year = {2021}
}
Comments
- The title is updated. - Definition 1.1 is fixed. - Several typos are fixed. - "the weak thresholding set" -> "the only weak thresholding set" (proof of Theorem 4.2). - An inaccuracy after equation (17) is corrected. - Some references are corrected, and some are added