论文标题
通过内部函数进行近似的扩张理论方法
A dilation theoretic approach to approximation by inner functions
论文作者
论文摘要
利用运算符理论在希尔伯特空间上的结果，我们证明了单位盘和单位bidisc上基质值霍明型函数的近似结果。 必不可少的工具是收缩的单一扩张理论和在$ h^\ inty $的单位球中实现功能的实现公式。我们首先证明了Carathéodory结果的概括。该概括具有许多应用程序。使用Potapov分解证明了矩阵值均连续延伸至单位圆的均匀近似结果。由于费舍尔，这概括了定理。 证明了矩阵值函数的近似结果，其天然相关的内核具有有限的负方块。这使用了Krein-anger-anger分解。 $ j $ - 收缩的meromorphic函数的近似结果，其中$ j $在$ \ mathbb c^n $上诱导了无限期的度量，使用Potapov-Ginzburg Theorem证明。此外，还证明了在某些其他感兴趣域中具有值的单位光盘上的全态函数的近似结果。
Using results from theory of operators on a Hilbert space, we prove approximation results for matrix-valued holomorphic functions on the unit disc and the unit bidisc. The essential tools are the theory of unitary dilation of a contraction and the realization formula for functions in the unit ball of $H^\infty$. We first prove a generalization of a result of Carathéodory. This generalization has many applications. A uniform approximation result for matrix-valued holomorphic functions which extend continuously to the unit circle is proved using the Potapov factorization. This generalizes a theorem due to Fisher. Approximation results are proved for matrix-valued functions for whom a naturally associated kernel has finitely many negative squares. This uses the Krein-Langer factorization. Approximation results for $J$-contractive meromorphic functions where $J$ induces an indefinite metric on $\mathbb C^N$ are proved using the Potapov-Ginzburg Theorem. Moreover, approximation results for holomorphic functions on the unit disc with values in certain other domains of interest are also proved.
