An inverse problem for weighted Paley-Wiener spaces

TL;DR

This paper addresses an inverse spectral problem for weighted Paley-Wiener spaces, reconstructing a Hamiltonian system from a measure that defines a spectral measure, under certain norm comparability conditions.

Contribution

It introduces a method to reconstruct the canonical Hamiltonian system from a measure related to weighted Paley-Wiener spaces, extending inverse spectral theory.

Findings

Reconstruction of Hamiltonian systems from spectral measures.

Conditions for norm comparability and density in $L^2( ext{measure})$.

Application to inverse problems in spectral theory.

Abstract

Let $\mu$ be a measure on the real line $\mathbb{R}$ such that $\int_{\mathbb{R}}\frac{d\mu(t)}{1+t^2} < \infty$ and let $a>0$. Assume that the norms $\|f\|_{L^2(\mathbb{R})}$ and $\|f\|_{L^2(\mu)}$ are comparable for functions $f$ in the Paley-Wiener space $PW_{a}$ and that $PW_a$ is dense in $L^2(\mu)$. We reconstruct the canonical Hamiltonian system $JX' = zHX$ such that $\mu$ is the spectral measure for this system.