On the Euler method of summability and concerning Tauberian theorems

Abstract

For any two regular summability methods (??) and (??), the condition under which ?? ? lim???? = ?? implies ?? ? lim???? = ?? is called a Tauberian condition and the corresponding theorem is called a Tauberian theorem. Usually in the theory of summability, the case in which the method ?? is equivalent to the ordinary convergence is taken into consideration. In this paper, we give new Tauberian conditions under which ordinary convergence or Cesàro summability of a sequence follows from its Euler summability by means of the product theorem of Knopp for the Euler and Cesàro summability methods.