On the rapidly convergence in capacity of the sequence of holomorphic functions

Abstract

We are interested in finding sufficient conditions on a Borel set $X$ lying either inside a bounded domain $D \subset \mathbb C^n$ or in the boundary $\partial D$ so that if $\{r_m\}_{m \ge 1}$ is a sequence of rational functions and $\{f_m\}_{m \ge 1}$ is a sequence of bounded  holomorphic functions on $D$ with $\{f_m-r_m\}_{m \ge 1}$ is convergent fast enough to $0$ in some sense on $X$ then the convergence occurs  on the whole domain $D$