  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 for all $z_0\in
\{z\in \mathbb{C}: \ |z|=1\}$, but $z_0=1$, the complex function
$(z^{1-\alpha}(z-1)^\alpha -T)^{-1}$ \ exists and is holomorphic in a
neighborhood of $z_0$, 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*} Three concrete examples are also included to illustrate the
obtained results.
