A Beurling-Carleson set which is a uniqueness set for a given weighted space of analytic functions期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

按检索

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

Authors:	Cyril Agrafeuil

Institution:	LaBAG, CNRS-UMR 5467, Université Bordeaux I, 351 cours de la Libération, 33451 Talence, France

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 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 to the unit circle $\mathbb{T}$ . Let be a closed subset of $\mathbb{T}$ . We say that 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 . In 1980, A. Atzmon asked whether there exists a sequence 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 , there exists a non-zero function in $B_{p}$ that vanishes on . In this note, we give a negative answer to this question.

Keywords:	Boundary zero of analytic functions sets of uniqueness spaces of analytic functions Beurling-Carleson sets

	点击此处可从《Proceedings of the American Mathematical Society》浏览原始摘要信息
	点击此处可从《Proceedings of the American Mathematical Society》下载免费的PDF全文

设为首页 | 免责声明 | 关于勤云 | 加入收藏