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 concept of spectrum sequences that are bounded given polynomial. We apply spectral theory to study asymptotic behavior solutions fractional linear difference equations. One obtained results is an extension famous Katznelson-Tzafriri Theorem, saying ?-resolvent associated with equation, satisfies estimate type, provided it polynomial, and equation on unit circle either empty or consists only one element 1. Three concrete examples also included illustrate results.