2022/07/28 by Geronimo, Jeffrey S., Iliev, Plamen
#Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2207.14383
We define a class of Bernstein-Szegő measures on ℝ2 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 determines a unique sequence of orthonormal polynomials which gives a simple formula for dμ/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 ℝ2. 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.