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A Beurling-Carleson set which is a uniqueness set for a given weighted space of analytic functions

Author: Cyril Agrafeuil
Journal: Proc. Amer. Math. Soc. 134 (2006), 3287-3294
MSC (2000): Primary 30C15, 30H05
Published electronically: May 8, 2006
MathSciNet review: 2231913
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Abstract: Let $ p = \big( p(n) \big)_{n \geq 0}$ be a sequence of positive real numbers. We define $ B_{p}$ as the space of functions $ f$ which are analytic in the unit disc $ \mathbb{D}$, continuous on $ \overline{\mathbb{D}}$ and such that

$\displaystyle \big\Vert f \big\Vert _{p} := \sum_{n=0}^{+\infty} \vert \hat{f}(n) \vert \, p(n) < +\infty, $

where $ \hat{f}(n)$ is the $ n^{\textrm{th}}$ Fourier coefficient of the restriction of $ f$ to the unit circle $ \mathbb{T}$. Let $ E$ be a closed subset of $ \mathbb{T}$. We say that $ E$ is a Beurling-Carleson set if

$\displaystyle \int_{0}^{2\pi} \log^{+} \frac{1}{d(e^{it},E)} \mathrm{d} t < +\infty, $

where $ d(e^{it},E)$ denotes the distance between $ e^{it}$ and $ E$. In 1980, A. Atzmon asked whether there exists a sequence $ p$ of positive real numbers such that $ \displaystyle \lim_{n \rightarrow +\infty} \frac{p(n)}{n^{k}} = +\infty$ for all $ k \geq 0$ and that has the following property: for every Beurling-Carleson set $ E$, there exists a non-zero function in $ B_{p}$ that vanishes on $ E$. In this note, we give a negative answer to this question.

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Cyril Agrafeuil
Affiliation: LaBAG, CNRS-UMR 5467, Université Bordeaux I, 351 cours de la Libération, 33451 Talence, France
Address at time of publication: LATP, Faculté des Sciences de Saint-Jérôme, Bâtiment Henri Poincaré, Cour A, 13397 Marseille cedex 20, France

Keywords: Boundary zero of analytic functions, sets of uniqueness, spaces of analytic functions, Beurling-Carleson sets
Received by editor(s): September 23, 2004
Received by editor(s) in revised form: May 26, 2005
Published electronically: May 8, 2006
Communicated by: David R. Larson
Article copyright: © Copyright 2006 American Mathematical Society