论文标题
磁盘中的汤姆森分解
Thomson decompositions of measures in the disk
论文作者
论文摘要
我们研究了确定$ p^2（μ）$的结构的经典问题，即在lebesgue空间中的分析多项式关闭$ l^2（μ）$的紧凑型borel borel量$μ$居住在复杂平面上。汤姆森（Thomson）在他的有影响力的作品中表明，该空间分解为完整的$ l^2 $空间和其他物品，这些空间本质上是飞机上域上分析功能的空间。 For a family of measures $μ$ supported on the closed unit disk $\overline{\mathbb{D}}$ which have a part on the open disk $\mathbb{D}$ which is similar to the Lebesgue area measure, and a part on the unit circle $\mathbb{T}$ which is the restriction of the Lebesgue linear measure to a general measurable subset $E$ of $ \ mathbb {t} $，我们扩展了khrushchev的想法，并计算了空间$ p^2（μ）$的汤姆森分解的确切形式。事实证明，我们介绍的$ \ mathbb {t} $的某些可测量子集根据某些可测量子集进行了分裂。我们强调了Cauchy积分操作员和De Branges-Rovnyak空间的应用。
We study the classical problem of identifying the structure of $P^2(μ)$, the closure of analytic polynomials in the Lebesgue space $L^2(μ)$ of a compactly supported Borel measure $μ$ living in the complex plane. In his influential work, Thomson showed that the space decomposes into a full $L^2$-space and other pieces which are essentially spaces of analytic functions on domains in the plane. For a family of measures $μ$ supported on the closed unit disk $\overline{\mathbb{D}}$ which have a part on the open disk $\mathbb{D}$ which is similar to the Lebesgue area measure, and a part on the unit circle $\mathbb{T}$ which is the restriction of the Lebesgue linear measure to a general measurable subset $E$ of $\mathbb{T}$, we extend the ideas of Khrushchev and calculate the exact form of the Thomson decomposition of the space $P^2(μ)$. It turns out that the space splits according to a certain decomposition of measurable subsets of $\mathbb{T}$ which we introduce. We highlight applications to the theory of the Cauchy integral operator and de Branges-Rovnyak spaces.
