Weighted-𝐿² polynomial approximation in ℂ

Abstract

We study the density of polynomials in H 2 ( Ω , e − φ ) H^2(\Omega ,e^{-\varphi }) , the space of square integrable holomorphic functions in a bounded domain Ω \Omega in C \mathbb {C} , where φ \varphi is a subharmonic function. In particular, we prove that the density holds in Carathéodory domains for any subharmonic function φ \varphi in a neighborhood of Ω ¯ \overline {\Omega } . In non-Carathéodory domains, we prove that the density depends on the weight function, giving examples.