Tauberian theorem for the Laplace-Stieltjes integral and Dirichlet series
type: Item
instance of: scholarly article
title: Tauberian theorem for the Laplace-Stieltjes integral and Dirichlet series
zbMATH Open document ID: 0712.40002
DOI: 10.1007/BF00971952
author: A. Mishkelyavichyus
published in: Lithuanian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: The author considers the functions f defined by the Laplace-Stieltjes integral \[ (1)\quad f(z)=\int^{\infty}_{0}e^{-\lambda (t)z} d\sigma (t) \] with complex-valued \(\lambda\) and \(\sigma\) satisfying a number of technical conditions. He proves 9 Tauberian theorems about this integral. The typical and the most interesting one is the following. Suppose \(z_ 0=x_ 0+iy_ 0\) belongs to the boundary of the region of convergence of (1) and let \(\eta (t)=\int^{t}_{0}e^{-\lambda (s)z_ 0} d\sigma (s).\) Suppose, moreover, that the limit \(\ell =\lim_{x\to x^+_ 0}f(x+iy_ 0)\) exists. Then \[ \int^{p}_{0}\lambda (t)d\eta (t)=o(\lambda (p)),\quad p\to \infty \] is the necessary and sufficient condition for the convergence of the integral (1) at \(z=z_ 0\) to the value \(\ell\).
reviewed by: J. Kaczorowski
Mathematics Subject Classification ID: 40E05
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11M45
Mathematics Subject Classification ID: 44A10
zbMATH DE Number: 4171593
zbMATH Keywords: Laplace-Stieltjes transform
zbMATH Keywords: Tauberian theorems
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4171593