Abstract
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 λ⩾1. We introduce the so-called λ-almost greedy and λ-partially greedy bases. The case when λ=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 λ-almost greedy for all (some) λ⩾1. However, for each λ>1, there exists an unconditional basis that is λ-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 λ-almost greedy with constant 1 for some λ>1, and(2)not strong partially greedy with constant 1 but is λ-partially greedy with constant 1 for some λ>1. Finally, we prove various characterizations of different greedy-type bases.
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