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Transactions of the American Mathematical Society

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Weighted-$L^2$ polynomial approximation in $\mathbb {C}$


Authors: Séverine Biard, John Erik Fornæss and Jujie Wu
Journal: Trans. Amer. Math. Soc. 373 (2020), 919-938
MSC (2010): Primary 30B60, 30E10; Secondary 32A10, 32E30, 32W05, 31A05
DOI: https://doi.org/10.1090/tran/7935
Published electronically: October 28, 2019
MathSciNet review: 4068254
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the density of polynomials in $H^2(\Omega ,e^{-\varphi })$, the space of square integrable holomorphic functions in a bounded domain $\Omega$ in $\mathbb {C}$, where $\varphi$ is a subharmonic function. In particular, we prove that the density holds in Carathéodory domains for any subharmonic function $\varphi$ in a neighborhood of $\overline {\Omega }$. In non-Carathéodory domains, we prove that the density depends on the weight function, giving examples.


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Additional Information

Séverine Biard
Affiliation: Science Institute, University of Iceland, Dunhagi 3, IS-107 Reykjavik, Iceland
Email: biard@hi.is

John Erik Fornæss
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, Sentralbygg 2, Alfred Getz vei 1, 7034 Trondheim, Norway
Address at time of publication: LAMAV, Université polytechnique Hauts-de-France, Campus du Mont d’Houy, 59313 Valenciennes Cedex 9, France
MR Author ID: 68145
Email: severine.biard@uphf.fr

Jujie Wu
Affiliation: School of Mathematics and Statistics, Henan University, Kaifeng 475001, Henan Province, People’s Republic of China; and Department of Mathematical Sciences, Norwegian University of Science and Technology, Sentralbygg 2, Alfred Getz vei 1, 7034 Trondheim, Norway
MR Author ID: 1249622
Email: jujie.wu@ntnu.no, 99jujiewu@tongji.edu.cn

Keywords: Carathéodory domain, weighted $L^2$-estimates, polynomial approximation, moon-shaped domain
Received by editor(s): June 19, 2018
Received by editor(s) in revised form: February 27, 2019
Published electronically: October 28, 2019
Additional Notes: The first author was supported in part by Rannis-grant 152572-051
The second author was supported by the Norwegian Research Council grant 240569
The third author was supported by the Norwegian Research Council grant 240569 and NSFC grant 11601120. The third author is the corresponding author
Article copyright: © Copyright 2019 American Mathematical Society