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

Let $X$ be a Banach space and $(e_n)_{n=1}^\infty$ be a basis. For a function $f$ in a large collection $\mathcal{F}$ (closed under composition), we define and characterize $f$-greedy and $f$-almost greedy bases. We study relations among these bases as $f$ varies and show that while a basis is not almost greedy, it can 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 $$f\mbox{-greedy}\ \Longleftrightarrow \ f\mbox{-almost greedy}.$$ We give various examples of Banach spaces to illustrate our results.