向量值分段连续几乎自守函数及其若干推论

Vector-valued almost automorphic distributions and vector-valued almost automorphic ultradistributions

In this paper, we introduce the notions of a vector-valued almost automorphic distribution and a vector-valued almost automorphic ultradistribution, working in the framework of complex Banach spaces. We prove several structural characterizations for the introduced classes.

Asymptotically almost periodic and asymptotically almost automorphic vector-valued generalized functions

The main purpose of this paper is to introduce the notion of an asymptotically almost periodic ultradistribution and asymptotically almost automorphic ultradistribution with values in a Banach space, as well as to further analyze the classes of asymptotically almost periodic and asymptotically almost automorphic distributions with values in a Banach space. We provide some applications of the introduced concepts in the analysis of systems of ordinary differential equations.

收缩算子的自守Nelson扩张与不变子空间跟踪

Automorphic Nelson Dilations for Contractions and Invariant Subspace Tracking

研究\(n\times n\)严格收缩矩阵\(T\)的自守Nelson扩张，通过该扩张可快速获其最小等距和酉扩张，与经典理论相连。还证明每个\(T\)的Nelson扩张有特别性质，如强纠缠特征值函数或与\(z\)无关的约化子空间，同时研究了相关矩阵乘积特征值情况。

具有分段连续符号的块Toeplitz行列式的渐近性

Asymptotics of block Toeplitz determinants with piecewise continuous symbols

本研究针对带有限跳跃的矩阵值分段连续符号，在温和条件下确定块Toeplitz行列式的渐近性，提出新的Toeplitz行列式局部化定理与Fredholm指标计算方法，结果可应用于量子自旋链模型纠缠熵研究。

Generalized vector valued almost periodic and ergodic distributions

For $\Cal A\subset L^1_{loc}(\Bbb J,X)$ let $\Cal M\Cal A$ consist of all $f\in L^1_{loc}$ with $ M_h f (\cdot):=\frac {1}{h}\int_{0}^{h}f(\cdot +s)\,ds \in \Cal A$ for all $h>0$. Here $X$ is a Banach space, $\Bbb J= (α,\infty), [α,\infty)$ or $\Bbb R$. Usually $\Cal A\subset\Cal M\Cal A\subset \Cal M^2\Cal A\subset ...$. The map $ \Cal A \to \Cal {D}'_{\Cal A}$ is iteration complete, that is $ \Cal {D}'_{\Cal {D}'_{\Cal A}}= \Cal {D}'_{\Cal A}$. Under suitable assumptions $\widetilde {\Cal M}^n \Cal {A}= \Cal A + \{T^{(n)} : T \in \Cal A\}$, and similarly for $\Cal {M}^n \Cal A$. Almost perio…

Spectral Asymptotics for Toeplitz Matrices Having Certain Piecewise Continuous Symbols

The limiting behavior of the eigenvalues of the Toeplitz matrices $T_{n}[σ]=(\hatσ(i-j))$, where $0\leq i,j \leq n$, as $n \to \infty$, is investigated in the case of complex valued functions $σ$ defined on the unit circle $\mathbb{T}$ and having exactly one point of discontinuity. It is found that if $σ(z)=(-z)^βτ(z)$, $β$ not an integer and $τ$ satisfying certain smoothness conditions, then $\det T_{n}[σ]=\mathbf{G}[τ]^{n+1}n^{-β^{2}}E[τ,β](1+o(1))$ as $n \to \infty$, where $\mathbf{G}[τ]$ denotes the geometric mean of $τ$ and $E$ is a constant independent of $n$. A value for $E$ is found in…