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arXiv · 1910.05777

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

Séverine Biard·John Erik Fornæss·Jujie Wu
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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

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Séverine Biard, John Erik Fornæss, Jujie Wu. 2019-10-13. Weighted $L^2$ version of Mergelyan and Carleman approximation. https://arxiv.org/abs/1910.05777

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