Before revision:  In this work, we provide a general framework of summarizing diagrams that we call Persistence Curves (PC). The main idea is so-called Fundamental Lemma of Persistent Homology, which is derived from the classic elder rule. Under this framework, certain  well-known summaries, such as persistent Betti numbers, and persistence landscape, are special cases of the PC . Moreover, we prove a rigorous bound for a general families of PCs.
After revision:  In this work, we provide a general framework of summarizing diagrams that we call the well-known summaries, such as persistent Betti numbers, and persistence landscape, are special cases of the PC . Moreover, we prove a rigorous bound for a general families of PCs.