Bernstein-Szegő measures in the plane
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866910100648624128 |
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| author | Geronimo, Jeffrey S. Iliev, Plamen |
| author_facet | Geronimo, Jeffrey S. Iliev, Plamen |
| contents | We define a class of Bernstein-Szegő 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 $μ$ on $\mathbb{R}$ 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 $\mathbb{R}^2$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_14383 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bernstein-Szegő measures in the plane Geronimo, Jeffrey S. Iliev, Plamen Classical Analysis and ODEs Complex Variables Functional Analysis We define a class of Bernstein-Szegő 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 $μ$ on $\mathbb{R}$ 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 $\mathbb{R}^2$. 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. |
| title | Bernstein-Szegő measures in the plane |
| topic | Classical Analysis and ODEs Complex Variables Functional Analysis |
| url | https://arxiv.org/abs/2207.14383 |