A time-varying Beurling-Lax theorem and a related interpolation problem (Q1842529)

This article concerns linear time-varying interpretations of the Beurling-Lax-Ball-Helton theorem and of Sarason's interpolation problem. The former characterizes shift-invariant \(H_ 2\) (Krein) subspace. Unilateral shift invariance reflects both causality and time invariance. Removing the stationarity requirement, a generalized theorem provides a characterization of certain causal subspace families \(M_ t \subset \mathcal{L}_ 2(t,\infty)\), \(t \in \mathbb{R}\). Sarason's interpolation problem is interpreted here as a search for a (close to) minimal induced norm system, given causal input-output specifications. The Beurling-Lax theorem helps in identifying admissible specification classes. The problem is then reduced to and solved in terms of a linear time-varying Nehari problem. Technically, developments are based on time-domain, state-space methods.