Geodesic
Weight-almost greedy bases
Informatics and Automation, Harmonic analysis, approximation theory, and number theory, Tome 303 (2018), pp. 120-141.

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We introduce the notion of a weight-almost greedy basis and show that a basis for a real Banach space is $w$-almost greedy if and only if it is both quasi-greedy and $w$-democratic. We also introduce the notion of a weight-semi-greedy basis and show that a $w$-almost greedy basis is $w$-semi-greedy and that the converse holds if the Banach space has finite cotype. 
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S. J. Dilworth; D. Kutzarova; V. N. Temlyakov; B. Wallis. Weight-almost greedy bases. Informatics and Automation, Harmonic analysis, approximation theory, and number theory, Tome 303 (2018), pp. 120-141. http://geodesic.mathdoc.fr/item/TRSPY_2018_303_a9/

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