The thresholding greedy algorithm versus approximations with sizes bounded by certain functions f - Chu, Hùng Việt (Journal of mathematical analysis and applications, 2024)
Let X be a Banach space and (en)n=1∞ be a basis. For a function f in a large collection F (closed under composition), we define and characterize f-greedy and f-almost greedy bases. We study relations among these bases as f varies and show that while a basis is not almost greedy, it can be f-greedy for some f∈F. Furthermore, we prove that for all non-identity function f∈F, we have the surprising equivalencef-greedy⟺f-almost greedy. We give various examples of Banach spaces to illustrate our results.