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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A Beurling-Carleson set which is a uniqueness set for a given weighted space of analytic functions
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by Cyril Agrafeuil PDF
Proc. Amer. Math. Soc. 134 (2006), 3287-3294 Request permission

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 \[ \big \| f \big \|_{p} := \sum _{n=0}^{+\infty } | \hat {f}(n) | p(n) < +\infty , \] 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 \[ \int _{0}^{2\pi } \log ^{+} \frac {1}{d(e^{it},E)} \mathrm {d} t < +\infty , \] 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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Additional Information
  • 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
  • Email: Cyril.Agrafeuil@math.u-bordeaux.fr
  • 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
  • © Copyright 2006 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 3287-3294
  • MSC (2000): Primary 30C15, 30H05
  • DOI: https://doi.org/10.1090/S0002-9939-06-08399-7
  • MathSciNet review: 2231913