Bernstein–Szegő measures in the plane

We define a class of Bernstein–Szegő measures on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi> μ </mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> determines a unique sequence of orthonormal polynomials which gives a simple formula for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d mu slash d x"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi> μ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">d\mu /dx</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi> μ </mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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.