Rational Approximation on Products of Planar Domains

Abstract

We consider $A(\Omega)$, the Banach space of functions $f$ from $ \overline{\Omega}=\prod_{i \in I} \overline{U_i}$ to $\mathbb{C}$ that are continuous with respect to the product topology and separately holomorphic, where $I$ is an arbitrary set and $U_i$ are planar domains of some type. We show that finite sums of finite products of rational functions of one variable with prescribed poles off $ \overline{U_i}$ are uniformly dense in $A(\Omega)$. This generalizes previous results where $U_i=\mathbb{D}$ is the open unit disc in $\mathbb{C}$ or $\overline{U_i}^c$ is connected.