Tauberian theorems for Abel limitability method
type: Item
instance of: scholarly article
title: Tauberian theorems for Abel limitability method
zbMATH Open document ID: 1162.40003
author: İbrahím Çanak
author: Umit Totur
published in: Central European Journal of Mathematics
publication date: 2008-06-25T00:00:00Z
review text: Suppose \(u=(u_n)\) is a sequence of real numbers and suppose that as \(n\to\infty\) we have \(u_n\to A(u)\). In this case \(u\) is also Abel summable to \(A(u)\), i.e., as \(x\to1\), we have \((1-x)\sum_{n=0}^{\infty}u_{n}x^n=A(u)\). The authors formulate new Tauberian conditions to prove converse statements. The theorems are formulated in terms of the operators \(I,S\) and \(T\) defined by \((Iu)_n=u_n\), \((Su)_n=\sum_{i=0}^n u_{i}/(n+1)\) and \((Tu)_n=n(u_n-u_{n-1})\) and generalise results of \textit{A. Tauber} [Monatsh. f. Math. 8, 273--277 (1897; JFM 28.0221.02)]. In the main result of the paper the authors prove that if for some integer \(m\geq1\) the sequence \((I-S)^{m+1}u\) is slowly oscillating, then Abel summability of \(u\) to \(A(u)\) implies that \(u_n\to A(u)\).
reviewed by: Edward Omey
Mathematics Subject Classification ID: 40E05
Mathematics Subject Classification ID: 40G10
Mathematics Subject Classification ID: 26A12
zbMATH DE Number: 5292963
zbMATH Keywords: Abel summability
zbMATH Keywords: Tauberian conditions
zbMATH Keywords: slow oscillation
MaRDI profile type: Publication
full work available at URL: s11533-008-0019-7
OpenAlex ID: W2024938065
cites work: A Tauberian theorem with a generalized one-sided condition
cites work: Q4815780
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cites work: Tauberian retrieval theory
DOI: 10.2478/S11533-008-0019-7
description: scientific article; zbMATH DE number 5292963