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 $\Vert f{\Vert }_{L^2(\mathbb{R})}$ and $\Vert f{\Vert }_{L^2(\mu )}$ are comparable for functions $f$ in the Paley-Wiener space $P{W}_a$ and that $P{W}_a$ is dense in $L^2(\mu )$. We reconstruct the canonical Hamiltonian system $J{X}^{\prime }=zHX$ such that $\mu$ is the spectral measure for this system.

14 pages

References in corpus (4)

• Inverse spectral problems for Sturm-Liouville operators with singular potentials
• Inverse spectral problems for Dirac operators with summable potentials
• Canonical systems and de Branges spaces
• Continuous analogs of polynomials orthogonal on the unit circle. Krein systems

Cited by in corpus (2)

• Periodic approximations in inverse spectral problems for canonical Hamiltonian systems
• Sampling measures, Muckenhoupt Hamiltonians, and triangular factorization