Homogeneous polynomials on strictly convex domains
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Abstract:
We consider a circular, bounded, strictly convex domain $\Omega \subset \mathbb C^{d}$ with boundary of class $C^{2}$. For any compact subset $K$ of $\partial \Omega$ we construct a sequence of homogeneous polynomials on $\Omega$ which are big at each point of $K$. As an application for any $E\subset \partial \Omega$ circular subset of type $G_{\delta }$ we construct a holomorphic function $f$ which is square integrable on $\Omega \setminus \mathbb DE$ and such that $E=E_{\Omega }^{2}(f):=\left \{z\in \partial \Omega : \int _{\mathbb Dz}\left |f\right |^{2}d\mathfrak {L}_{\mathbb Dz}^{2} =\infty \right \}$ where $\mathbb D$ denotes unit disc in $\mathbb C$.References
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Additional Information
- Piotr Kot
- Affiliation: Politechnika Krakowska, Instytut Matematyki, ul. Warszawska 24, 31-155 Kraków, Poland
- Email: pkot@usk.pk.edu.pl
- Received by editor(s): September 8, 2005
- Received by editor(s) in revised form: September 20, 2006
- Published electronically: September 10, 2007
- Communicated by: Mei-Chi Shaw
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 3895-3903
- MSC (2000): Primary 32A05, 32A40
- DOI: https://doi.org/10.1090/S0002-9939-07-08939-3
- MathSciNet review: 2341939