The Thresholding Greedy Algorithm versus Approximations with Sizes Bounded by Certain Functions $f$

Hung Viet Chu

Published 2022-09-20Version 1

Let $X$ be a Banach space and $(e_n)_{n=1}^\infty$ be a basis. For a fixed function $f$ in a certain collection $\mathcal{F}$ (closed under composition), we define and characterize ($f$, greedy) and ($f$, almost greedy) bases. These bases nontrivially extend the classical notion of greedy and almost greedy bases. We study relations among ($f$, (almost) greedy) bases as $f$ varies and show that while a basis is not almost greedy, it can still be ($f$, greedy) for some $f\in \mathcal{F}$. Furthermore, we prove that for all non-identity function $f\in \mathcal{F}$, we have the surprising equivalence $$\mbox{($f$, greedy)}\ \Longleftrightarrow \ \mbox{($f$, almost greedy)}.$$