On the distributional Stieltjes transformation
type: Item
instance of: scholarly article
title: On the distributional Stieltjes transformation
zbMATH Open document ID: 0611.46050
DOI: 10.1155/S0161171286000388
author: D. Nikolić-Despotović
author: Arpad Takači
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46141
review text: 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''.
Mathematics Subject Classification ID: 46F12
Mathematics Subject Classification ID: 44A15
zbMATH DE Number: 3987996
zbMATH Keywords: distributional Stieltjes transformation
zbMATH Keywords: Abelian theorems
zbMATH Keywords: quasiasymptotic behaviour at infinity
MaRDI profile type: Publication
OpenAlex ID: W2047244561
description: scientific article; zbMATH DE number 3987996