Notes on functions on the unit disk

A class of Bernstein-type operators on the unit disk

We construct and study sequences of linear operators of Bernstein-type acting on bivariate functions defined on the unit disk. To this end, we study Bernstein-type operators under a domain transformation, we analyse the bivariate Bernstein-Stancu operators, and we introduce Bernstein-type operators on disk quadrants by means of continuously differentiable transformations of the function. We state convergence results for continuous functions and we estimate the rate of convergence. Finally some interesting numerical examples are given, comparing approximations using the shifted Bernstein-Stancu…

单位圆盘内偏theta函数的零点不存在性

No zeros of the partial theta function in the unit disk

该研究证明了当参数q在(-1,0)∪(0,1)区间时，偏theta函数在闭单位圆盘内不存在零点，完善了偏theta函数的零点分布理论。

Analytic Differential Operators on the Unit Disk

Formally symmetric differential operators on weighted Hardy-Hilbert spaces are analyzed, along with adjoint pairs of differential operators. Eigenvalue problems for such operators are rather special, but include many of the classical Riemann and Heun equations. Symmetric minimal operators are characterized. A regular class whose leading coefficients have no zeros on the unit circle are shown to be essentially self-adjoint. Eigenvalue asymptotics are established. Some extensions to non-self-adjoint operators are also considered.

Wiener's Tauberian theorem in L^1(G//K) and harmonic functions in the unit disk

Our main result is to give necessary and sufficient conditions, in terms of Fourier transforms, on a closed ideal $I$ in $\loneg$, the space of radial integrable functions on $G=SU(1,1)$, so that $I=\loneg$ or $I=\lonez$---the ideal of $\loneg$ functions whose integral is zero. This is then used to prove a generalization of Furstenberg's theorem which characterizes harmonic functions on the unit disk by a mean value property and a ``two circles" Morera type theorem (earlier announced by Agranovski\uı).

关于论文“极多项式在单位圆盘上的零点定位”的注记

A Note on the Paper "Localization of Zeros of Polar Polynomials on the Unit Disk"

该注记指出Costas-Santos等人的论文核心结论不成立，修正后建立了带精确界的Grace–Szego定理并刻画卷积方法扩展条件，纠正了已发表记录。

几乎保持双曲面积的单位圆盘解析映射

Analytic mappings of the unit disk which almost preserve hyperbolic area

本文研究对大双曲圆盘面积扭曲有界的单位圆盘解析自映射，通过角导数、Lipschitz扩张、Möbius扭曲、临界点分布及Aleksandrov-Clark测度等对其进行刻画，并探讨了相关测度的Lyapunov指数。