Krein-type theorems and ordered structure for Cauchy-de Branges spaces

Cauchy-de Branges spaces, geometry of their reproducing kernels and multiplication operators

Cauchy-de Branges spaces are Hilbert spaces of entire functions defined in terms of Cauchy transforms of discrete measures on the plane and generalizing the classical de Branges theory. We consider extensions of two important properties of de Branges spaces to this, more general, setting. First, we discuss geometric properties (completeness, Riesz bases) of systems of reproducing kernels corresponding to the zeros of certain entire functions associated to the space. In the case of de Branges spaces they correspond to orthogonal bases of reproducing kernels. The second theme of the paper is a c…

Localization of zeros in Cauchy-de Branges spaces

We study the class of discrete measures in the complex plain with the following property: up to a finite number, all zeros of any Cauchy transform of the measure (with $\ell^2$-data) are localized near the support of the measure. We find several equivalent forms of this property and prove that the parts of the support attracting zeros of Cauchy transforms are ordered by inclusion modulo finite sets.

De Branges' theorem on approximation problems of Bernstein type

The Bernstein approximation problem is to determine whether or not the space of all polynomials is dense in a given weighted $C_0$-space on the real line. A theorem of L. de Branges characterizes non--density by existence of an entire function of Krein class being related with the weight in a certain way. An analogous result holds true for weighted sup--norm approximation by entire functions of exponential type at most $τ$ and bounded on the real axis ($τ>0$ fixed). We consider approximation in weighted $C_0$-spaces by functions belonging to a prescribed subspace of entire functions which is s…

移位算子的循环性与de Branges-Rovnyak空间中的一个相关完备性问题

Cyclicity of the shift operator and a related completeness problem in de Branges-Rovnyak spaces

该文研究de Branges-Rovnyak空间上移位算子的循环向量，通过柯西变换建立其与完备性问题的关联，进而得到循环性相关的若干推论。

规范化 Cauchy 变换的 Fourier 系数

Fourier coefficients of normalized Cauchy transforms

本文研究规范化 Cauchy 变换 Fourier 系数的唯一性问题，通过同时逼近与循环内函数存在性分析，给出 Kahane–Katzenelson 定理的新非概率证明，并证明 de Branges–Rovnyak 空间中不存在一致容许控制函数。

作为de Branges-Rovnyak空间的无限支撑调和加权Dirichlet空间

Infinitely supported harmonically weighted Dirichlet spaces which are de Branges Rovnyak spaces

本文研究调和加权Dirichlet空间与de Branges-Rovnyak空间的相等问题，突破有限支撑限制，刻画了无限支撑测度下两类空间相等的条件，还建立了相关T(1)型结果与扰动定理，并证明了Brown-Shields猜想。