Weighted $L^2$ version of Mergelyan and Carleman approximation

Abstract

We study the density of polynomials in $H^2(E,\phi )$, the space of square integrable functions with respect to $e^{-\phi }dm$ and holomorphic on the interior of $E$ in $\mathbb{C}$, where $\phi$ is a subharmonic function and $dm$ is a measure on $E$. We give a result where $E$ is the union of a Lipschitz graph and a Carathéodory domain, that we state as a weighted $L^2$-version of the Mergelyan theorem. We also prove a weighted $L^2$-version of the Carleman theorem. Keywords: Mergelyan theorem, Carleman theorem, Weighted $L^2$- spaces, Rectifiable non-Lipschitz arc

26 pages, Comments are welcome