论文标题
伯恩斯坦 - 西格赛在飞机上的测量
Bernstein-Szegő measures in the plane
论文作者
论文摘要
我们为$ \ mathbb {r}^2 $定义了一类Bernstein-Szegő衡量标准，并建立了它们的频谱属性，从而提供了一维理论的自然扩展。我们还得出了涉及有限多个时刻的条件，这些时刻在二维环境中是新的，并且完全表征了这些措施。本文的重要部分致力于用于基质值功能的伯恩斯坦 - 塞格理论的独立发展。我们还展示了双变量三角多项式的Fejér-riesz分解问题的最新结果如何用于构建与$ \ Mathbb {r}^2 $相关的与Bernstein-Szegő测量相关的空间的显式基础。这些证明使用了来自真实分析，复杂分析和代数的技术混合物。
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. An important part of the paper is devoted to a self-contained development of the Bernstein-Szegő theory for matrix-valued functionals. We also show how recent results in the Fejér-Riesz factorization problem for bivariate trigonometric polynomials can be used to construct explicit bases of the spaces associated with the Bernstein-Szegő measures on $\mathbb{R}^2$. The proofs use a mixture of techniques from real analysis, complex analysis and algebra.
