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

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 {fm}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 χm(|f − fm|2) converges to 0 in capacity on X. We are interested in finding conditions (on \(\mathcal A\)) so that every sequence {rm}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.