A revisited Tauberian theorem for which slow decrease with respect to a weight function is a tauberian condition for the weighted mean summability of integrals over R+

In this extended abstract, we present an alternative proof of a Tauberian theorem of slowly decreasing type with respect to the weight function due to Karamata [5] for the weighted mean summable real-valued integrals over R+ := [0, ∞).Some particular choices of weight functions provide alternative proofs of some well-known Tauberian theorems given for several important summability methods.