Geodesic
Weight-almost greedy bases
Trudy Matematicheskogo Instituta imeni V.A. Steklova, 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. 
@article{TM_2018_303_a9,
     author = {S. J. Dilworth and D. Kutzarova and V. N. Temlyakov and B. Wallis},
     title = {Weight-almost greedy bases},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {120--141},
     publisher = {mathdoc},
     volume = {303},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2018_303_a9/}
}
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S. J. Dilworth; D. Kutzarova; V. N. Temlyakov; B. Wallis. Weight-almost greedy bases. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Harmonic analysis, approximation theory, and number theory, Tome 303 (2018), pp. 120-141. http://geodesic.mathdoc.fr/item/TM_2018_303_a9/