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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 defineand characterize $f$-greedy and $f$-almost greedy bases. We study relationsamong these bases as $f$ varies and show that while a basis is not almostgreedy, it can be $f$-greedy for some $f\in \mathcal{F}$. Furthermore, we provethat for all non-identity function $f\in \mathcal{F}$, we have the surprisingequivalence $$f\mbox{-greedy}\ \Longleftrightarrow \ f\mbox{-almost greedy}.$$We give various examples of Banach spaces to illustrate our results.

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