An alternative proof of a Tauberian theorem for the weighted mean summability of integrals over R_{+}
Articles > Article · 2020
Authors: CAGLA KAMBAK, İbrahim ÇANAK
Creative Mathematics and Informatics, International, Scopus, 2020, vol 29, pp 45-50
Summary
Let 0 ≡ q(x) be a nondecreasing function on R + := [0, ∞) such that q(0) = 0 andfor every ρ > 1.Given a real-or complex-valued function f ∈ L 1 loc (R + ), we define s(x) := x 0 f (u)du and its weighted mean asIf the limit limx→∞ s(x) = l exists, then limx→∞ σ(s(x)) = l also exists.However, the converse is not true in general.The converse implication might hold under some appropriate conditions.These conditions are called Tauberian conditions and theorems involving these conditions are called Tauberian theorems.In this paper, we have given an alternative proof of the Tauberian theorem stating that the existence of limx→∞ s(x) = l follows from that of limx→∞ σ(s(x)) = l and a Tauberian condition.If f (x) is a real or complex-valued function, then this Tauberian condition is one-sided or two-sided, respectively.