论文标题
多项式界限序列和应用于离散系统渐近行为的光谱理论
A Spectral Theory of Polynomially Bounded Sequences and Applications to the Asymptotic Behavior of Discrete Systems
论文作者
论文摘要
在本文使用翻译操作员定义的转换中，我们介绍了由$ n^ν$界定的序列频谱的概念，其中$ν$是自然的数字。我们将这种光谱理论应用于研究$δ^αx（n）= tx（n）+y（n）$，$ n \ in \ mathbb {n} $的分数差方程解的渐近行为，其中$ 0 <α\ le 1 $。获得的结果之一是著名的katznelson-tzafriri定理的扩展，说，如果$α$ -Resolvent操作员$s_α$满足$ \ sup_ {n \ in \ mathbb {n}}} \ | s_α（n）\ | /n^ν<\ infty $，对于所有$ z_0 \ in \ {z \ in \ Mathbb {c}：\ | z | | = 1 \} $，但是$ z_0 = 1 $，复杂函数$（z^{1-α}（z-1）（z-1）（z-1）^α-t） \ begin {align*} \ lim_ {n \ to \ infty} \ frac {1} {n^νz} \ sum_ {k = 0}^{ν+1} \ frac {（ν+1）！} = 0。 \ end {Align*}还包括三个具体示例，以说明获得的结果。
In this paper using a transform defined by the translation operator we introduce the concept of spectrum of sequences that are bounded by $n^ν$, where $ν$ is a natural number. We apply this spectral theory to study the asymptotic behavior of solutions of fractional difference equations of the form $Δ^αx(n)=Tx(n)+y(n)$, $n\in \mathbb{N}$, where $0<α\le 1$. One of the obtained results is an extension of a famous Katznelson-Tzafriri Theorem, saying that if the $α$-resolvent operator $S_α$ satisfies $\sup_{n\in\mathbb{N}} \| S_α(n)\| /n^ν<\infty$ and for all $z_0\in \{z\in \mathbb{C}: \ |z|=1\}$, but $z_0=1$, the complex function $(z^{1-α}(z-1)^α-T)^{-1}$ \ exists and is holomorphic in a neighborhood of $z_0$, then \begin{align*} \lim_{n\to \infty} \frac{1}{n^ν} \sum_{k=0}^{ν+1} \frac{(ν+1)!}{k!(ν+1-k)!} (-1)^{ν+1+k} S_α(n+k) =0. \end{align*} Three concrete examples are also included to illustrate the obtained results.
