Weak greedy algorithms and the equivalence between semi-greedy and almost greedy Markushevich bases. (English) Zbl 1528.41093
Summary: 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 infinite dimensional Banach spaces, the notions of semi-greedy, branch semi-greedy, weak semi-greedy, and almost greedy Markushevich bases are all equivalent. This completes and extends some results from [P. M. Berná, J. Math. Anal. Appl. 470, No. 1, 218–225 (2019; Zbl 1457.46015); S. J. Dilworth et al., Stud. Math. 159, No. 1, 67–101 (2003; Zbl 1056.46014); S. J. Dilworth et al., J. Funct. Anal. 263, No. 12, 3900–3921 (2012; Zbl 1267.46026)]. 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.
MSC:
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
| 46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |
| 46B20 | Geometry and structure of normed linear spaces |
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