On the distributional Stieltjes transformation
This paper is concerned with some general theorems on the distributional Stieltjes transformation. Some Abelian theorems are provided. Abelian Theorem: Let \(T\in {\mathcal S}_+'\) have quasiasymptotic behaviour at infinity of order a related to a regularly varying function \(r(t)=t^ aL(t)\). Then \[ (i)\quad T\in {\mathcal I}'(z)\quad for\quad z>\max (-1,a) \] and \[ (ii)\quad S_ z\{T\}(t)\sim \frac{\Gamma (z-a)}{\Gamma (z+1)}L(s)s^{a-z}\quad as\quad s\to \infty. \] Remark. Such a statement was proved by J. Lavoine and O. P. Misra for \(r(t)=t^ a\), \(a>-1\) and for \(r(t)=t^ a\log^ jt\), \(a>-1.\) In this paper there is a trivial example (it can be generalized easily) which shows again that quasiasymptotic behaviour is more appropriate for final value type Abelian theorems for Stieltjes transformation than equivalence at infinity, though the latter seems more ''natural''.
scientific article; zbMATH DE number 3987996