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

TL;DR

This paper extends classical approximation theorems to weighted $L^2$ spaces of holomorphic functions, establishing density of polynomials on complex sets combining Lipschitz graphs and Carathéodory domains.

Contribution

It introduces weighted $L^2$-versions of Mergelyan and Carleman theorems for complex sets with Lipschitz and non-Lipschitz boundaries.

Findings

Polynomials are dense in the specified weighted $L^2$ spaces on complex sets.

Weighted approximation results hold for unions of Lipschitz graphs and Carathéodory domains.

Theorems generalize classical approximation results to weighted $L^2$ contexts.

Abstract

We study the density of polynomials in $H^2(E,\varphi)$, the space of square integrable functions with respect to $e^{-\varphi}dm$ and holomorphic on the interior of $E$ in $\mathbb{C}$, where $\varphi$ 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 Carath\'{e}odory 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