Weighted \(L^2\) version of Mergelyan and Carleman approximation (Q2024695)

It is proved that the polynomials are dense in \(H^2(E,\varphi)\), the space of functions holomorphic in \(\mathrm{int}(E)\) and square integrable over \(E\subset\mathbb{C}\) with respect to the weighted measure \(e^{-\varphi}dm\) where \(\varphi\) is subharmonic and \(m\) a measure on \(E\). In the authors' previous paper [Trans. Am. Math. Soc. 373, No. 2, 919--938 (2020; Zbl 1439.30007)] this was proved for a Carthéodory domain \(E=\Omega\subset\mathbb{C}\). Here it is proved for \(E=\gamma\), a Lipschitz graph, and, under certain conditions, when \(E\) is the union of Carathéodory domains and Lipschitz arcs. This also results in a general Carleman-type theorem: for \(E=\Gamma\subset\mathbb{C}\), a locally Lipschitz continuous graph over \(\mathbb{R}\), then \(f\in L^2(\Gamma,\varphi)\) can be approximated arbitrary well by an entire function on every subgraph \(\Gamma_n\) that corresponds to the the interval \([n,n+1]\subset\mathbb{R}\), \(n\in\mathbb{Z}\).