We study the density of polynomials in \(H^2(E,\varphi )\), the space of square integrable functions with respect to \(\mathrm{e}^{-\varphi }\mathrm{d}m\) 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 Carathéodory domain, which we state as a weighted \(L^2\)-version of the Mergelyan theorem. We also prove a weighted \(L^2\)-version of the Carleman theorem.

中文翻译：

---

Mergelyan和Carleman逼近的加权$$ L ^ 2 $$ L2版本

我们研究\（H ^ 2（E，\ varphi）\）中多项式的密度，平方可积函数相对于\（\ mathrm {e} ^ {-\ varphi} \ mathrm {d} m \ ）并在\（{\ mathbb {C}} \）中E的内部全纯，其中\（\ varphi \）是次谐波函数，d m是对E的度量。我们给出一个结果，其中E是Lipschitz图和Carathéodory域的并集，我们将其表示为Mergelyan定理的加权\（L ^ 2 \） -版本。我们还证明了Carleman定理的加权\（L ^ 2 \）-版本。