Weighted-$L^2$ polynomial approximation in $\mathbb{C}$

Abstract

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

23 pages. Comments are welcome!