Enlarge Greedy Sums in Greedy-Type Properties by Different Factors

Abstract

It was previously known that the almost greedy (AG) property essentially remains the same when we enlarge greedy sums in the classical definition by a factor $\hat{I}»⩾1$. The present paper shows that if instead, we enlarge greedy sums in a reformulation of the AG property, we obtain a weaker one. However, the new property is essentially independent of the enlarging factor $\hat{I}»$ once $\hat{I}»>1$. In contrast, we observe a continuum of partially greedy-like properties by varying $\hat{I}»\in [1,\infty )$. Last but not least, under a threshold for $\hat{I}»$, we characterize the isometric version of the weakened AG property. Specifically, the characterization holds if and only if $\hat{I}»\in [1,2]$.

18 pages