Bauer-type factorization of positive matrices and the theory of matrix polynomials orthogonal on the unit circle
Bauer-type factorization of positive matrices and the theory of matrix polynomials orthogonal on the unit circle
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正矩阵的Bauer型因式分解及单位圆上正交矩阵多项式理论
DOI：
10.1109/tcs.1978.1084443
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发表时间：
1978
期刊：
影响因子：
--
通讯作者：
Nerses N. Kazanjian
中科院分区：
文献类型：
--
作者：
D. Youla;Nerses N. Kazanjian
本文表明，Bauer 对单位圆上非负标量多项式进行 Wiener-Hopf 分解的技术可以扩展到满足 Paley-Wiener 准则的任意可积周期 n × n 非负定 Hermitian 矩阵 K(\theta)。这是最通用的可能设置。由此产生的算法与 Rissanen 和 Kailath 最近导出的算法一致，但以基本方式建立，没有施加任何不必要的约束。该方法还提供了一些有关收敛性质的详细信息。分析的一个重要副产品是阐明了由权重 K(θ) 生成的两组矩阵正交多项式在谱分解中所起的作用。这些多项式可以递归地生成，并且对其限制性质的研究表明，它们为构造所需的维纳-霍普夫因子提供了有效的替代方案。由于矩阵K(θ)不限于某个有理矩阵的边界值，因此该算法还可以用于解决以Wiener-Hopf思想为中心的许多不同类型的电磁场问题。
In this paper it is shown that a technique due to Bauer for the Wiener-Hopf factorization of scalar polynomials that are nonnegative on the unit circle, can be extended to arbitrary integrable periodic n \times n nonnegative-definite Hermitian matrices K(\theta) which satisfy the Paley-Wiener criterion. This is the most general possible setting. The resulting algorithm agrees with the one derived recently by Rissanen and Kailath but is established in an elementary manner without the imposition of any unnecessary constraints. The method also supplies some detailed information regarding the nature of the convergence. An important byproduct of the analysis is the clarification of the role played in spectral factorization by two sets of matrix orthogonal polynomials generated by the weight K(\theta) . These polynomials can be generated recursively and a study of their limiting properties reveals that they provide an effective alternative scheme for the construction of the desired Wiener-Hopf factor. Since the matrix K(\theta) is not restricted to be the boundary value of some rational matrix, the algorithm can also be employed in the solution of many different types of electromagnetic field problems centered around the Wiener-Hopf idea.