Tauberian-type theorems with application to the Stieltjes transformation

The Abelian and Tauberian-type theorems were introduced by Stanković [7] and Pilipović et al. [5]. In the first part of this paper, we give the definition of the quasiasymptotic expansion at 0+ and the quasiasymptotic behaviour of distributions at infinity from S′+ introduced in [1]. In this paper, we give the definition of space L′(r), classical Stieltjes transformation, modified Stieltjes transformation, and generalized modified Stieltjes transformation. This enables us to obtain, in the second part of the paper, the Tauberiantype theorems of quasiasymptotic behaviour of distributions at infinity.We give sufficient conditions under which the behaviour at infinity of the modified Stieltjes transformation Γ(r +1)(Tr+1 f )(x), r ∈R\(−N), f ∈ L′(r) determines the quasiasymptotic behaviour of f at infinity.