Interpretation of the Schur-Cohn test in terms of canonical systems

TL;DR

The paper develops a constructive bridge between the classical Schur–Cohn test for exponential polynomials and the theory of (quasi) canonical systems by interpreting the Schur–Cohn determinants as determinants of Hamiltonians in a de Branges framework. It provides an explicit, finite-dimensional construction of a piecewise-constant Hamiltonian $H(t)$ from exponential-polynomial data, establishes the corresponding quasi-canonical system, and connects root-distribution within the unit circle to the sign changes of ${ m Tr}(H_1, frac{}{} frac{}{}{ m Tr}H_d)$. The main contributions include a detailed inverse problem correspondence between polynomials with a given number of roots on the unit circle and sequences of $2 imes2$ real symmetric matrices in ${ m SL}_2(b R)$, an explicit direct-problem representation of the solution via 2-by-2 transfer matrices, and a generalized sufficiency criterion extending prior results. Together, these results deepen the link between Schur–Cohn theory and canonical-system methods, with potential for explicit Hamiltonian-driven reconstruction of polynomials from prescribed root distributions.

Abstract

We solve direct and inverse problems for two-dimensional (quasi) canonical systems related to exponential polynomials of a specific but sufficiently general type. The approach to the inverse problem in this paper provides an interpretation of the matrices and their determinants in the classical Schur-Cohn test for polynomials in terms of Hamiltonians of canonical system.

Theorems & Definitions (40)

• Theorem 1.1
• Remark 1.1
• Corollary 1.2
• Theorem 1.2
• Remark 1.3
• Theorem 1.3
• Corollary 1.4
• Lemma 3.1
• proof
• Lemma 3.2