Tauberian type gap conditions for Cesàro summation methods

Abstract

In this paper, we study gap conditions ensuring, for the class of series satisfying them, that the Cesaro methods (C, k) and (C, α) with fixedk and α (0<k<α) are equivalent. We obtain a sufficient condition in which gaps of the corresponding sizes separate uniformly bounded sets ofk+1 nonzero elements rather than neighboring nonzero elements (as in the classical case ofk=0).