On non-separated zero sequences of solutions of a linear differential equation

Operator Diagonalizations of Multiplier Sequences

We consider hyperbolicity preserving operators with respect to a new linear operator representation on $\mathbb{R}[x]$. In essence, we demonstrate that every Hermite and Laguerre multiplier sequence can be diagonalized into a sum of hyperbolicity preserving operators, where each of the summands forms a classical multiplier sequence. Interestingly, this does not work for other orthogonal bases; for example, this property fails for the Legendre basis. We establish many new formulas concerning the $Q_k$'s of Peetre's 1959 differential representation for linear operators in the specific case of He…

关于函数序列及其微分多项式的一个正规定则

A Normal Criterion Concerning Sequence of Functions and their Differential Polynomials

本文证明了开单位圆盘内亚纯函数序列的一个正规定则：在极点重数、非零性及微分多项式与扰动全纯函数差值零点数受限条件下，该函数序列正规。

有限阶微分算子序列的超循环子空间

Hypercyclic subspaces for sequences of finite order differential operators

该文研究复系数多项式序列对应的有限阶微分算子序列的超循环子空间问题，证明存在满足非零元均超循环的无限维闭子空间与稠密𝔠维子空间，且二者均可包含任意指定超循环函数。

整函数空间上微分算子的频繁超循环序列

Frequently hypercyclic sequences of differential operators on the space of entire functions

本文提出了整函数空间上卷积算子序列频繁超循环性的一个判据，该判据要求算子的生成函数在适当的环域上非零，且序列各项的模在边界上受前后项控制。

广义矩序列及其在若干矩偏微分方程中的应用

Sequences of generalized moments and their applications to some moment partial differential equations

本文引入广义矩序列，推广Balser矩可和性理论，构造Laplace和Borel算子扩展，刻画矩偏微分方程可和解及其奇异方向跳跃，以超函数和核函数表达。

Diagonal Differential Operators

We explore differential operators, $T$, that diagonalize on a simple basis, $\{B_n(x)\}_{n=0}^\infty$, with respect to some sequence of real numbers, $\{a_n\}_{n=0}^\infty$, and sequence of polynomials, $\{Q_k(x)\}_{k=0}^\infty$, as in $ T[B_n(x)]:=\left(\sum_{k=0}^\infty Q_k(x) D^k\right)B_n(x)=a_n B_n(x)$ for every $n\in\mathbb{N}_0$. We discover new relationships between the sequence, $\{Q_k(x)\}_{k=0}^\infty$, and the sequence, $\{a_n\}_{n=0}^\infty$. We find new relationships between polynomial interpolated eigenvalues and the sequence, $\{°(Q_k(x))\}_{k=0}^\infty$.

Polya sequences, Toeplitz kernels and gap theorems

A separated sequence $Λ$ on the real line is called a Polya sequence if any entire function of zero exponential type bounded on $Λ$ is constant. In this paper we solve the problem by Polya and Levinson that asks for a description of Polya sets. We also show that the Polya-Levinson problem is equivalent to a version of the so-called Beurling gap problem on Fourier transforms of measures. The solution is obtained via a recently developed approach based on the use of Toeplitz kernels and de Branges spaces of entire functions.

分离行列式点过程与广义Fock空间

Separated determinantal point processes and generalized Fock spaces

本文刻画了由倍增次调和权定义的广义Fock空间所对应行列式点过程几乎必然分离的条件，并通过与同强度Poisson过程比较揭示内在排斥力的作用，证明典范权情形下分离的临界指数为α<4/3。

Equivalence of summatory conditions along sequences for bounded holomorphic functions

A sequence of points $z_k$ in the unit disk is said to be thin for a given decrease function $ρ$, if there is a nontrivial bounded holomorphic function such that the infinite series $\sum_k ρ(1-|z_k|)|f(z_k)|$ converges. All sequences will be assumed hyperbolically separated. We give necessary and sufficient conditions for the problem of thinness of a sequence to be non-trivial (one way or the other), and for two different decrease functions to give rise to the same thin sequences. Along the way, some concrete conditions (necessary or sufficient) for a sequence to be thin are obtained.

Interpolating Sequences for Weighted Bergman Spaces on Strongly Pseudoconvex Bounded Domains

Let $0<p<\infty$, $β>-1$, and $Ω$ be a strongly pseudoconvex bounded domain with a smooth boundary in $\mathbb{C}^n$. We will study the interpolation problem for weighted Bergman spaces $A^p_β(Ω)$. In the case, $1\leq p<\infty$, and $β> \max \{n(2p-1)-1, n(2q-1)-1\}$, where $q$ is the conjugate exponent of $p$ (let $q=1$, for $p=1$), we show that a sequence in $\mathbb{B}_n$, the unit ball in $\mathbb{C}^n$, is interpolating for $A^p_β(\mathbb{B}_n)$ if and only if it is separated.

零点位于预先给定序列上的复微分方程的解

Solutions of complex differential equation having zeros on pre-given sequences

该复分析研究探讨了单位圆盘内特定解析系数下二阶线性微分方程解的零点序列性质，解答了公开问题，刻画了Carleson测度条件下零点序列的等价特征，并改进了经典的单叶函数相关结果。

Estimates of formal solutions for some generalized moment partial differential equations

Using increasing sequences of real numbers, we generalize the idea of formal moment differentiation first introduced by W. Balser and M. Yoshino. Slight departure from the concept of Gevrey sequences enables us to include a wide variety of operators in our study. Basing our approach on tools such as the Newton polygon and divergent formal norms, we obtain estimates for formal solutions of certain families of generalized linear moment partial differential equations with constant and time variable coefficients.