Performance of the Thresholding Greedy Algorithm with Larger Greedy Sums

The goal of this paper is to study the performance of the Thresholding Greedy Algorithm (TGA) when we increase the size of greedy sums by a constant factor $\lambda\geqslant 1$. We introduce the so-called $\lambda$-almost greedy and $\lambda$-partially greedy bases. The case when $\lambda = 1$ gives us the classical definitions of almost greedy and (strong) partially greedy bases. We show that a basis is almost greedy if and only if it is $\lambda$-almost greedy for all (some) $\lambda \geqslant 1$. However, for each $\lambda > 1$, there exists an unconditional basis that is $\lambda$-partially greedy but is not $1$-partially greedy. Furthermore, we investigate and give examples when a basis is 1. not almost greedy with constant $1$ but is $\lambda$-almost greedy with constant $1$ for some $\lambda > 1$, and 2. not strong partially greedy with constant $1$ but is $\lambda$-partially greedy with constant $1$ for some $\lambda > 1$. Finally, we prove various characterizations of different greedy-type bases.