Lower-Order Refinements of Greedy Approximation

Abstract

For two countable ordinals $α$ and $β$, a basis of a Banach space $X$ is said to be $(α, β)$-quasi-greedy if it is 1) quasi-greedy, 2) $\mathcal{S}_α$-unconditional but not $\mathcal{S}_{α+1}$-unconditional, and 3) $\mathcal{S}_β$-democratic but not $\mathcal{S}_{β+1}$-democratic. If $α$ or $β$ is replaced with $\infty$, then the basis is required to be unconditonal or democratic, respectively. Previous work constructed a $(0,0)$-quasi-greedy basis, an $(α, \infty)$-quasi-greedy basis, and an $(\infty, α)$-quasi-greedy basis. In this paper, we construct $(α, β)$-quasi-greedy bases for $β\le α+1$ (except the already solved case $α= β= 0$).