On the convergence of weighted mean summable improper integrals over R?0

Abstract

Given a real-valued function a(x) which is locally integrable in the sense of Lebesgue Rover R->= 0, we obtain Tauberian conditions for the convergence of the integral integral(infinity)(0)a(t)dt out of the weighted mean summability. Furthermore, we discuss summability of improper integrals via iterations of weighted means and provide corresponding Tauberian theorems.