A Spectral Theory of Polynomially Bounded Sequences and Applications to the Asymptotic Behavior of Discrete Systems

Abstract

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.