Bernstein-Szeg\H{o} measures in the plane

Abstract

We define a class of Bernstein-Szeg\H{o} measures on $\mathbb{R}^2$ and we establish their spectral properties, providing a natural extension of the one-dimensional theory. We also derive conditions involving finitely many moments, which are new in the two-dimensional setting, and which completely characterize these measures. A key ingredient in the theory on the real line stems from the fact that a measure $\mu$ on $\mathbb{R}$ determines a unique sequence of orthonormal polynomials which gives a simple formula for $d\mu/dx $ in the Bernstein-Szeg\H{o} family. Since there is no canonical way to introduce orthonormal polynomials in the plane, our extension is based on a new identity which connects a Fej\'er-Riesz factorization of the weight to a polynomial depending on three variables associated with $\mu$. Using recent results in the bivariate trigonometric Fej\'er-Riesz factorization problem, we define a nontrivial two-dimensional extension of the Szeg\H{o} mapping which provides explicit orthonormal bases of the spaces associated with Bernstein-Szeg\H{o} measures on $\mathbb{R}^2$. An important part of the paper is devoted to a self-contained development of the Bernstein-Szeg\H{o} theory for matrix-valued functionals. The proofs combine techniques from real analysis, complex analysis and algebra.