Isometric Theory for Greedy-Type Properties with Enlarged Greedy Sums

For a semi-normalized Markushevich basis, we study isometric almost-greedy inequalities in which a greedy sum of $ϕ(m)$ terms is compared with coordinate projections onto at most $m$ terms. Here $ϕ$ is any strictly increasing map of the nonnegative integers into themselves with $ϕ(0)=0$. We sandwich $ϕ$-almost greediness with constant $1$ ($ϕ$-AG$_1$) between two levels of symmetry for largest coefficients (SLC): $$ϕ\mbox{-SLC}_1\ \Longrightarrow\ ϕ\mbox{-AG}_1\ \Longrightarrow\ \mbox{weak }ϕ\mbox{-SLC}_1.$$ Both implications are equivalences if and only if $ϕ(1)\le2$. We then characterize $ϕ$-AG$_1$ by the weighted weak $ϕ$-SLC or a single homogeneous norm inequality, called Property $(F_ϕ)$.