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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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2018, Vol. 303, pp. 120–141.
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Dilworth, S.J., Kutzarova, D., Temlyakov, V. et al. Weight-Almost Greedy Bases. Proc. Steklov Inst. Math. 303, 109–128 (2018). https://doi.org/10.1134/S0081543818080102