Bauer-type factorization of positive matrices and the theory of matrix polynomials orthogonal on the unit circle

Abstract

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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n \times n</tex> nonnegative-definite Hermitian matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K(\theta)</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K(\theta)</tex> . 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K(\theta)</tex> 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.