Convergence of Sequences of Rational Functions on $\mathbb {C}^{n}$

Abstract

Let $\mathcal A:=\{\chi _{m}\}_{m \ge 1}$ be a sequence of continuous, real valued functions defined on $[0, \infty )$. We say that a sequence {f _m } _{m ≥ 1} of measurable functions defined on $X \subset D \subset \mathbb C^{n}$ is convergent in capacity (relative to D) with respect to the weight sequence $\mathcal A$ to a function f if χ _m (|f − f _m |²) converges to 0 in capacity on X. We are interested in finding conditions (on $\mathcal A$) so that every sequence {r _m } _{m ≥ 1} of rational functions on $\mathbb {C}^{n}$ converges in capacity with respect to $\mathcal A$ to a holomorphic function f defined on a bounded domain $D \subset \mathbb {C}^{n}$ provided that the convergence holds only pointwise on a small subset of D.