Brune sections in the nonstationary case
 Citation data:

Linear Algebra and its Applications, ISSN: 00243795, Vol: 343, Page: 389418
 Publication Year:
 2002
 Repository URL:
 https://digitalcommons.chapman.edu/scs_articles/462
 DOI:
 10.1016/s00243795(01)004190
 Author(s):
 Publisher(s):
 Tags:
 Mathematics; Nonstationary linear systems; Boundary interpolation; Brune sections; Timevarying systems; Algebra; Discrete Mathematics and Combinatorics; Other Mathematics
article description
Rational J innervalued functions which are J inner with respect to the unit circle ( J being a matrix which is both selfadjoint and unitary) play an important role in interpolation theory and are extensively utilized in signal processing for filtering purposes and in control for minimal sensitivity ( H ∞ feedback). Any such function is a product of three kinds of elementary factors, each of them having a unique singularity outside the unit disk, inside the unit disk and on the unit circle, respectively. Counterparts of the first kind have already been studied in the context of nonstationary systems, when analytic functions are replaced by upper triangular operators. The purpose of the present work is to study the nonstationary analogues of the factors of the third kind. One main difficulty is that one leaves the realm of bounded upper triangular operators and considers unbounded operators. Yet, as is the case for a number of special clases of nonstationary systems, all the systems under consideration are finitely specified, and the computations are done recursively on a finite set of state space data. We consider the particular case, where the operator given is of the IVI type (that is, it is timeinvariant both for small and large indices, and is timevarying in between). The theory results in a rather general factorization theorem that generalizes the timeinvariant case to finitely specified, timevarying systems.