2013

On the Euler method of summability and concerning Tauberian theorems

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.

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