We extend the well-known Katznelson-Tzafriri theorem, originally posed for power-bounded operators, to the case of Cesàro bounded operators of any order alpha > 0. For this purpose, we use a functional calculus between a new class of fractional Wiener algebras and the algebra of bounded linear operators, defined for operators with the corresponding Cesàro boundedness. Finally, we apply the main theorem to get ergodicity results for the Cesàro means of bounded operators
