On the Euler method of summability and concerning Tauberian theorems

Abstract

For any two regular summability methods (𝑈) and (𝑉), the condition under which 𝑉 − lim𝑥𝑛 = 𝜆 implies 𝑈 − lim𝑥𝑛 = 𝜆 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 𝑈 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.