Asymptotic equivalence and summability
type: Item
instance of: scholarly article
title: Asymptotic equivalence and summability
zbMATH Open document ID: 0788.40001
DOI: 10.1155/S0161171293000948
author: Mousa S. Marouf
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1994-05-19T00:00:00Z
full work available at URL: https://eudml.org/doc/46934
review text: This paper studies the relationships between the asymptotic equivalence of two sequences \((\lim_ n x_ n/y_ n=1\); i.e. \(x\sim y)\) and three variations for this equivalence. Let \(A\) be a nonnegative sequence-to- sequence transformation matrix, \(R_ m Az:= \sum_{n\geq m} |(Az)_ n|\), \(S_ m Az:= \sum_{n\leq m} | (Az)_ n|\), and \(\mu_ m Az:=\sup_{n\geq m} | (Az)_ n|\). The three variations are given by \(R_ m Ax/R_ m Ay\), \(S_ m Ax/S_ m Ay\), and \(\mu_ m Ax/\mu_ m Ay\). A typical result is the following. If \(x\) and \(y\) are nonvanishing null sequences such that \(x\sim y\), then \(\mu x\sim\mu y\). A study of these relationships when the limit is zero was done by \textit{J. A. Fridy} [Can. J. Math. 30, 808-816 (1978; Zbl 0359.40003)].
Mathematics Subject Classification ID: 40C05
zbMATH DE Number: 438929
zbMATH Keywords: weighted mean matrices
zbMATH Keywords: asymptotic equivalence
reviewed by: Billy E. Rhoades
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 438929