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arXiv · 2105.15177

Bidemocratic bases and their connections with other greedy-type bases

Fernando Albiac·Jose L. Ansorena·Miguel Berasategui·Pablo M. Berna·Silvia Lassalle
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Abstract

In nonlinear greedy approximation theory, bidemocratic bases have traditionally played the role of dualizing democratic, greedy, quasi-greedy, or almost greedy bases. In this article we shift the viewpoint and study them for their own sake, just as we would with any other kind of greedy-type bases. In particular we show that bidemocratic bases need not be quasi-greedy, despite the fact that they retain a strong unconditionality flavor which brings them very close to being quasi-greedy. Our constructive approach gives that for each $1<p<\infty$ the space $\ell_p$ has a bidemocratic basis which is not quasi-greedy. We also present a novel method for constructing conditional quasi-greedy bases which are bidemocratic, and provide a characterization of bidemocratic bases in terms of the new concepts of truncation quasi-greediness and partially democratic bases.

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Fernando Albiac, Jose L. Ansorena, Miguel Berasategui, Pablo M. Berna, Silvia Lassalle. 2021-05-31. Bidemocratic bases and their connections with other greedy-type bases. https://arxiv.org/abs/2105.15177

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