Let X be a Banach space and (enβ)n=1ββ be a basis. For a fixed
function f in a certain collection 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 fβF. Furthermore, we prove that for all non-identity
function fβF, we have the surprising equivalence \mbox{($f$,
greedy)}\ \Longleftrightarrow \ \mbox{($f$, almost greedy)}.Comment: 20 page