arXiv 2101.023062021-01-07math.CVmath.FA

从圆盘到四块的全纯映射插值

Interpolation by holomorphic maps from the disc to the tetrablock

Hadi O. Alshammari, Zinaida A. Lykova

备注：: 35 pages, the paper has been accepted for publication in the Journal of Mathematical Analysis and Applications

摘要：

四嵌段是集合$$ \mathcal{E}=\{x \in \mathbb{C}^3: \quad 1-x_1z-x_2w+x_3z w \neq 0 \quad whenever \quad |z|\leq 1, |w|\leq 1\}. $$，$\mathcal{E}$的闭包用$\overline{\mathcal{E}}$表示。四内函数是从单位圆$ \mathbb{D} $到$\overline{\mathcal{E}}$的解析映射$x$，使得对于单位圆$ \mathbb{T}$的几乎所有点$λ$，\[ \lim_{r\uparrow 1} x(r λ) \mbox{ exists and lies in } b \overline{\mathcal{E}}, \]，其中$b \overline{\mathcal{E}}$表示$\overline{\mathcal{E}}$的区别边界。有一个有理四内函数$ x$的度的自然概念;它仅仅是从$ \mathbb{T} $到$ b \overline{\mathcal{E}} $的连续映射$ x|_\mathbb{T} $的拓扑度。本文给出了构造一般有理四内函数$n$的一个处方。处方利用已知的构造有限Blaschke产品的给定程度，满足一定的插值条件的帮助下，从插值数据形成的Pick矩阵。已知如果$x= (x_1, x_2, x_3)$是次数为$n$的有理四元内函数，则$x_1 x_2 - x_3$要么是相同的$0$，要么在闭单位圆$\overline{\mathbb{D}}$中精确地具有$n$零，用重数计数。事实证明，一个自然的选择数据建设的合理四内函数$x= (x_1, x_2, x_3)$包括点在$\overline{\mathbb{D}}$为其中$x_1 x_2 - x_3=0$和价值观的$x$在这些点。

英文摘要：

The tetrablock is the set $$ \mathcal{E}=\{x \in \mathbb{C}^3: \quad 1-x_1z-x_2w+x_3z w \neq 0 \quad whenever \quad |z|\leq 1, |w|\leq 1\}. $$ The closure of $\mathcal{E}$ is denoted by $\overline{\mathcal{E}}$. A tetra-inner function is an analytic map $x$ from the unit disc $ \mathbb{D} $ to $\overline{\mathcal{E}}$ such that, for almost all points $λ$ of the unit circle $ \mathbb{T}$, \[ \lim_{r\uparrow 1} x(r λ) \mbox{ exists and lies in } b \overline{\mathcal{E}}, \] where $b \overline{\mathcal{E}}$ denotes the distinguished boundary of $\overline{\mathcal{E}}$. There is a natural notion of degree of a rational tetra-inner function $ x$; it is simply the topological degree of the continuous map $ x|_\mathbb{T} $ from $ \mathbb{T} $ to $ b \overline{\mathcal{E}} $. In this paper we give a prescription for the construction of a general rational tetra-inner function of degree $n$. The prescription exploits a known construction of the finite Blaschke products of given degree which satisfy some interpolation conditions with the aid of a Pick matrix formed from the interpolation data. It is known that if $x= (x_1, x_2, x_3)$ is a rational tetra-inner function of degree $n$, then $x_1 x_2 - x_3$ either is identically $0$ or has precisely $n$ zeros in the closed unit disc $\overline{\mathbb{D}}$, counted with multiplicity. It turns out that a natural choice of data for the construction of a rational tetra-inner function $x= (x_1, x_2, x_3)$ consists of the points in $\overline{\mathbb{D}}$ for which $x_1 x_2 - x_3=0$ and the values of $x$ at these points.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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