One-sided tauberian conditions for the (N p) summability of integrals
Articles > Article · 2018
Authors: Ümit Totur, Muhammet Ali Okur, İbrahim ÇANAK
UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, Journal Indexed in SCI Expanded, International, Scopus, 2018, vol 80, pp 65-74
Summary
Let p be a function on ℝ+ := [0, ∞) which is integrable in Lebesgue's sense over every finite interval (0, x) for 0 < x < ∞, in symbol: p ϵ L1loc(ℝ+) such that P(x) = fx0 p(t)dt ≠ 0 for each x > 0 P(0) = 0 and P(x) → ∞ as x → ∞. For a real-valued function f ϵ L1loc(ℝ+), we set s(x) := fx0 f(t)dt and σ(1)p (x) := 1/P(x)fx0 s(t)p(t)dt, x > 0, provided that P(x) > 0. We say that f∞0 f(x)dx is summable by the weighted mean method determined by the function P(x) if there exists some s ϵ ℝ such that (equation presented) If the limit (equation presented) exists, then so does (equation presented) In this paper, we obtain some new Tauberian conditions in terms of the weighted general control modulo for the weighted mean method of integrals in order that the converse implication hold true. Our results generalize some classical type Tauberian theorems given for Cesaro summability method of integrals.