Four dimensional matrix characterizations of the Pringsheim core
type: Item
instance of: scholarly article
title: Four dimensional matrix characterizations of the Pringsheim core
zbMATH Open document ID: 1054.40006
author: Richard F. Patterson
published in: Southeast Asian Bulletin of Mathematics
publication date: 2004-11-25T00:00:00Z
review text: In an earlier work [Int.~J.~Math.~Math.~Sci.~ 22, 785--793 (1999; Zbl 0949.40007)], the author defined the Pringsheim core, \(P-C \{x\}\), of a double sequence \([x]\), to be \(\bigcap^\infty_{n=1} P-C_n \{x\}\), where \(P-C_n\{x\}\) is the least, closed convex set that includes all points \(x_{kl}\), with both \(k\) and \(l\) greater than \(n\), and proved the following analog of the Knopp core theorem:\ If \(A\) is a nonnegative, four-dimensional, RH-regular summability matrix, then \(P-C\{Ax\}\subseteq P-C\{x\}\), for every bounded sequence for which \([Ax]\) exists. In the present article, the author defines absolute equivalency of two RH-regular matrices, for a given class of double sequences, gives necessary and sufficient conditions for RH-regular matrices to be absolutely equivalent, for all bounded sequences, and, finally, establishes the following result: The Pringsheim core of \(A\{s\}\), where \([s]\) is a bounded (double) sequence, is contained in the core of \([s]\) if and only if \(A\) is RH-regular and is absolutely equivalent to a nonnegative matrix \(B\), for all bounded sequences.
Mathematics Subject Classification ID: 40B05
zbMATH DE Number: 2119030
zbMATH Keywords: Knopp core theorem
zbMATH Keywords: summability matrix
zbMATH Keywords: absolute equivalency
zbMATH Keywords: double sequences
zbMATH Keywords: Prings\-heim core
reviewed by: Robert E. Zink
MaRDI profile type: Publication
Recommended article: Double sequence core theorems
Recommended article: The Cesàro core of double sequences
Recommended article: The \(\sigma\)-convergence and \(\sigma\)-core of double sequences
Recommended article: Analogues of some fundamental theorems of summability theory
Recommended article: Uniformly summable double sequences
Recommended article: The Riesz convergence and Riesz core of double sequences
Recommended article: Matrix transformations of double sequences
Recommended article: Q5403261
Recommended article: Some matrix transformations and generalized core of double sequences
Recommended article: Equivalence of methods for the summation of double sequences
description: scientific article; zbMATH DE number 2119030