Lower-Order Refinements of Greedy Approximation

For two countable ordinals $\alpha$ and $\beta$, a basis of a Banach space $X$ is said to be $(\alpha, \beta)$-quasi-greedy if it is 1) quasi-greedy, 2) $\mathcal{S}_\alpha$-unconditional but not $\mathcal{S}_{\alpha+1}$-unconditional, and 3) $\mathcal{S}_\beta$-democratic but not $\mathcal{S}_{\beta+1}$-democratic. If $\alpha$ or $\beta$ 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 $(\alpha, \infty)$-quasi-greedy basis, and an $(\infty, \alpha)$-quasi-greedy basis. In this paper, we construct $(\alpha, \beta)$-quasi-greedy bases for $\beta\le \alpha+1$ (except the already solved case $\alpha = \beta = 0$).