Bidemocratic bases and their connections with other greedy-type bases

TL;DR

This paper explores bidemocratic bases in nonlinear greedy approximation, showing they can lack quasi-greediness, and introduces new methods and characterizations for these bases, expanding understanding beyond their traditional dual roles.

Contribution

It demonstrates that bidemocratic bases need not be quasi-greedy and introduces novel construction and characterization techniques for these bases.

Findings

Bidemocratic bases can be non-quasi-greedy.

For each 1<p<∞, ℓ_p has a bidemocratic basis that is not quasi-greedy.

New concepts of truncation quasi-greediness and partial democracy characterize bidemocratic bases.

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.