Suppose we have three pairwise disjoint infinite sets $A,B,C$ which are subsets of $[0,1]$.
It can be assumed that $A,B,C$ are countable. Also, each sum converges absolutely.
Their elements are unknown.
Given three functions $f_A,f_B,f_C:[0,1]\to \mathbb R$, we know that for every integer $n\ge 0$, $$\sum_{p\in A} f_A(p)~p^{n} +\sum_{p\in B}f_B(p)~p^n+n\sum_{p\in C}f_{C}(p)p^{n-1}=C_n$$ where $C_1,C_2,\cdots$ are known.
Under what conditions can we obtain a unique solution of the elements in $A,B,C$?
In general, when does a system of infinite equations (and thus infinite unknowns) has a unique solution? Moreover, is there a branch of mathematics that specifically studies this topic?