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

In this paper using a transform defined by the translation operator we introduce the concept of spectrum of sequences that are bounded by $n^\nu$, where $\nu$ is a natural number. We apply this spectral theory to study the asymptotic behavior of solutions of fractional difference equations of the form $\Delta^\alpha x(n)=Tx(n)+y(n)$, $n\in \mathbb{N}$, where $0<\alpha\le 1$. One of the obtained results is an extension of a famous Katznelson-Tzafriri Theorem, saying that if the $\alpha$-resolvent operator $S_\alpha$ satisfies $\sup_{n\in\mathbb{N}} \| S_\alpha (n)\| /n^\nu <\infty$ and the set of $z_0\in \mathbb{C}$ such that $(z-\tilde k^\alpha (z)T)^{-1}$ exists, and together with $\tilde k^\alpha (z)$, is holomorphic in a neighborhood of $z_0$ consists of at most $1$, where $ \tilde k^\alpha (z)$ is the Z-transform of $k^\alpha (n):= \Gamma (\alpha +n)/(\Gamma (\alpha )\Gamma (n+1))$, then \begin{align*} \lim_{n\to \infty} \frac{1}{n^\nu} \sum_{k=0}^{\nu+1} \frac{(\nu+1)!}{k!(\nu+1-k)!} (-1)^{\nu+1+k} S_\alpha (n+k) =0. \end{align*}