Sampling measures, Muckenhoupt Hamiltonians, and triangular factorization

Let $\mu$ be an even measure on the real line $\mathbb{R}$ such that $$c_1{\int }_{\mathbb{R}}\mathrm{\mid }f{\mathrm{\mid }}^{2}dx\le {\int }_{\mathbb{R}}\mathrm{\mid }f{\mathrm{\mid }}^{2}d\mu \le c_2{\int }_{\mathbb{R}}\mathrm{\mid }f{\mathrm{\mid }}^{2}dx$$ for all functions $f$ in the Paley-Wiener space ${\mathrm{P}\mathrm{W}}_a$. 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.