An inverse problem for weighted Paley-Wiener spaces

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.