Lower-Order Refinements of Greedy Approximation

Abstract

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

28 pages, 1 figure. To appear in Studia Mathematica