On the construction of de Branges spaces for dynamical systems associated with finite Jacobi matrices

TL;DR

The paper constructs a de Branges space attached to a dynamical system driven by a finite Jacobi matrix $A$ using the Boundary Control Method. It derives the forward problem, documents the spectral data $\\{\\lambda_k,\\rho_k\\}$, and defines the connecting operator $C^T$ so that the reachable set yields a finite-dimensional Fourier image $B_N$. This image is proven to be a de Branges space with reproducing kernel $J_z$ built from a Hermite-Biehler function $E$, thereby linking boundary data to analytic function spaces. The approach suggests potential extensions to multidimensional discrete systems and provides a framework for inverse spectral analysis in this discrete setting.

Abstract

We consider dynamical systems with boundary control associated with finite Jacobi matrices. Using the method previously developed by the authors, we associate with these systems special Hilbert spaces of analytic functions (de Branges spaces)

Theorems & Definitions (6)

• Lemma 1
• Lemma 2
• Theorem 1
• Remark 1
• Theorem 2
• Theorem 3