Conditions under which the convergence of a sequence or its certain subsequences follows from the summability by deferred weighted means

Abstract

UDC 517.5 Let ( u k ) be a sequence of real or complex numbers. First, we consider a real sequence ( u k ) and formulate one-sided Tauberian conditions, which are necessary and sufficient for the convergence of certain subsequences of ( u k ) to follow from its deferred weighted summability. These conditions are satisfied if ( u k ) is deferred slowly decreasing or if ( u k ) obeys a Landau-type Tauberian condition. Second, we consider a complex sequence ( u k ) and present a two-sided Tauberian condition which is necessary and sufficient in order that the convergence of certain subsequences of ( u k ) follow from its deferred weighted summability. This condition is satisfied either if ( u k ) is deferred slowly oscillating or if ( u k ) obeys a Hardy-type Tauberian condition. Finally, we extend these results to sequences in ordered linear spaces over the real numbers.