Rational approximation on $A^{\infty }(\varOmega )$
Volume 248 / 2019
Abstract
Let $\{U_i\}_{i\in I}$ be a family of bounded planar domains in $\mathbb C$. Here $A(\prod_{i\in I}U_i)$ stands for the set of continuous functions on $\prod_{i\in I}\overline{U_i}$ endowed with the product topology that are separately holomorphic on each $U_i$. We consider the algebra $A^\infty(\prod_{i\in I}U_i)$ consisting of those functions in $A(\prod_{i\in I}U_i)$ all of whose mixed partial derivatives are also in $A(\prod_{i\in I}U_i)$. We show that under certain conditions the set of functions that are finite sums of finite products of rational functions of one variable whose poles are prescribed elements of $\overline{U}_i^c$ is dense in $A^\infty(\prod_{i\in I}U_i)$ with the topology of uniform convergence for every finite set of mixed partial derivatives.
