Sampling measures, Muckenhoupt Hamiltonians, and triangular factorization

Abstract

Let $\mu$ be an even measure on the real line $\mathbb{R}$ such that $$c_1{\int }_{\mathbb{R}}|f{|}^{2}dx\le {\int }_{\mathbb{R}}|f{|}^{2}d\mu \le c_2{\int }_{\mathbb{R}}|f{|}^{2}dx$$ for all functions $f$ in the Paley-Wiener space ${\mathrm{PW}}_a$. We prove that $\mu$ is the spectral measure for the unique Hamiltonian $\mathcal{H}=\left(w&00&\frac{1}{w}\right)$ on $[0,a]$ generated by a weight $w$ from the Muckenhoupt class $A_2[0,a]$. As a consequence of this result, we construct Krein's orthogonal entire functions with respect to $\mu$ and prove that every positive, bounded, invertible Wiener-Hopf operator on $[0,a]$ with real symbol admits triangular factorization.

19 pages

References in corpus (2)

• Canonical systems and de Branges spaces
• An inverse problem for weighted Paley-Wiener spaces

Cited by in corpus (1)

• On recurrence coefficients of Steklov measures