Bernstein–Szegő measures in the plane

We define a class of Bernstein–Szegő measures on R 2 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 μ on R R determines a unique sequence of orthonormal polynomials which gives a simple formula for d μ / d x d /dx in the Bernstein–Szegő 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 Fejér–Riesz factorization of the weight to a polynomial depending on three variables associated with μ. Using recent results in the bivariate trigonometric Fejér–Riesz factorization problem, we define a nontrivial two-dimensional extension of the Szegő mapping which provides explicit orthonormal bases of the spaces associated with Bernstein–Szegő measures on R 2 R^2. An important part of the paper is devoted to a self-contained development of the Bernstein–Szegő theory for matrix-valued functionals. The proofs combine techniques from real analysis, complex analysis and algebra.