Let $ f : \mathbb{C} \to \mathbb{C}$ be a continuous function with $ f(0)=0$ .

Let $ \{a_i\}_{i\in \mathbb{N}}$ be a set of scalars in $ \mathbb{C}$ such that $ $ \exists C > 0 : \forall i\in \mathbb{N} : |a_i| \leq C $ $ Can we always find a holomorphic function $ g$ on $ B(0,C+1)$ (the open disk of radius $ C+1$ ) such that $ g(0)=0$ and $ \sum_{i \in \mathbb{N}} |(f-g)(a_i)|^2 < +\infty$ ?