Performance of the Thresholding Greedy Algorithm with Larger Greedy Sums

The goal of this note is to study the performance of the Thresholding Greedy Algorithm (TGA) when we increase the size of greedy sums by a constant factor $\lambda\ge 1$. We introduce the so-called ($\lambda$, partially greedy) bases. While the case $\lambda = 1$ gives strong partially greedy bases, we show that, for each $\lambda > 1$, there exists a (Schauder) basis that is ($\lambda$, partially greedy) but is not strong partially greedy. Furthermore, we investigate and give examples when a basis is 1. not $1$-(almost) greedy but the TGA still gives the smallest error from an $m$-term approximation if we allow greedy sums to be of size $\lceil \lambda m\rceil$, and 2. not $1$-strong partially greedy but $1$-($\lambda$, partially greedy) for some $\lambda > 1$. Finally, we prove various equivalences for different greedy-type bases.