I wonder if there is a name for a family $F$ of meromorphic functions $f : \mathbb C \to \mathbb C$ all of whose poles are contained in a fixed closed discrete set $S = (s_i)_{i \in \mathbb N}$ with orders of the poles $s_i$ bounded by $(n_i)_{i \in \mathbb N}$ independently of $f$.
Equivalently:
- there exists a fixed holomorphic function $g$ such that all $fg$ are holomorphic.
- they are all sections of some fixed sheaf $O(D)$ of meromorphic functions, for some divisor $D$
- with the holomorphic ringed-space structure of $\mathbb C$, the smallest $\mathcal O_{\mathbb C}$-module with the $f \in F$ as global sections is locally of finite type.
I'm thinking uniformly meromorphic but am curious if there already exists a name, or if there is a related term from algebraic geometry which I can borrow.
This shows up, for example, when working with meromorphic functions that take values in a (Fréchet) function space; each meromorphic $F : \mathbb C \to C^\infty(\mathbb R)$ gives such a family, for example.