Abstract
In this paper using a transform defined by the translation operator we introduce the concept of spectrum of sequences that are bounded by a given polynomial. We apply this spectral theory to study the asymptotic behavior of solutions of fractional linear difference equations. One of the obtained results is an extension of a famous Katznelson-Tzafriri Theorem, saying that the α-resolvent operator that is associated with the fractional equation, satisfies an asymptotic estimate of Katznelson-Tzafriri type, provided that it is bounded by the polynomial, and the spectrum of the fractional equation on the unit circle is either empty or consists of only one element 1. Three concrete examples are also included to illustrate the obtained results.
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