Tauberian theorems for iterations of weighted mean summable integrals

Abstract

Let p be a positive weight function on which is integrable in Lebesgue's sense over every finite interval in symbol: such that for each and For a real- valued function and denote. But the converse of this implication is not true in general. In this paper, we obtain some Tauberian theorems for the weighted mean method of integrals in order that the converse implication holds true. Our results extend and generalize some classical type Tauberian theorems given for Cesaro and logarithmic summability methods of integrals. we say that iteration of weighted mean method determined by the function integrable to a finite number L and we write s(the existence of the limit limx.8 But the converse of this implication is not true in general. In this paper, we obtain some Tauberian theorems for the weighted mean method of integrals in order that the converse implication holds true. Our results extend and generalize some classical type Tauberian theorems given for Cesaro and logarithmic summability methods of integrals.