Sampling measures, Muckenhoupt Hamiltonians, and triangular factorization

TL;DR

This paper establishes a connection between measures satisfying certain sampling inequalities, Muckenhoupt Hamiltonians, and the triangular factorization of Wiener-Hopf operators, providing new insights into spectral measures and operator factorizations.

Contribution

It proves that measures satisfying sampling bounds are spectral measures for specific Muckenhoupt Hamiltonians and demonstrates that positive Wiener-Hopf operators with real symbols admit triangular factorization.

Findings

Measures satisfying sampling inequalities correspond to spectral measures of Muckenhoupt Hamiltonians.

Constructs Krein's orthogonal entire functions with respect to these measures.

Shows that certain Wiener-Hopf operators admit 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.