Brune Sections in the Nonstationary Case
Rational J-inner-valued functions which are J-inner with respect to the unit circle (J being a matrix which is both self-adjoint 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 non-stationary systems, when analytic functions are replaced by upper triangular operators. The purpose of the present work is to study the non-stationary 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 non-stationary 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 time-invariant both for small and large indices, and is time-varying in between). The theory results in a rather general factorization theorem that generalizes the time-invariant case to finitely specified, time-varying systems.
D. Alpay, V. Bolotnikov, P. Dewilde and A. Dijksma. Brune sections in the nonstationary case. Linear Algebra and Applications, 343-344 (2002), 389-418.