Double sequence core theorems
type: Item
instance of: scholarly article
title: Double sequence core theorems
zbMATH Open document ID: 0949.40007
DOI: 10.1155/S0161171299227858
author: Richard F. Patterson
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2000-12-03T00:00:00Z
full work available at URL: https://eudml.org/doc/48573
review text: Let \(s=\{s_n\}\) denote a complex sequence. Define \(R_n\) to be the smallest closed convex region of the complex plane which contains \(\{s_n, s_{n+1}, \dots\}\), \(R:=\bigcap^\infty_{n=1} R_n\). The set \(R\) is called the core of \(s\). Knopp's core theorem states that, if \(A\) is a nonnegative regular matrix, then the core of the \(A\)-transform of \(s\) is contained in the core of \(s\). For double sequences, the usual convergence used is that of Pringsheim, which leads to the Toeplitz conditions of regularity of a four-dimensional matrix (written RH-regular), for convergent sequences which are also bounded. In this paper the author defines the concept of a core sequence for double sequences and proves the two-dimensional analog of Knopp's core theorem. The notation \(P\)-limit means the limit in the Pringsheim sense. The author also proves that, for all real-valued double sequences \([x]\), the condition \(P\)-\(\lim \sup[Ax] \leq P\)-\(\lim\sup [x]\) is equivalent to \(A\) being an RH-regular matrix satisfying \(P\)-\(\lim_{m,n} \sum^\infty_{k,j=0} |a_{mnjk} |=1\).
Mathematics Subject Classification ID: 40B05
Mathematics Subject Classification ID: 40C05
zbMATH DE Number: 1417977
zbMATH Keywords: Pringsheim convergence
zbMATH Keywords: complex sequence
zbMATH Keywords: Knopp's core theorem
zbMATH Keywords: convergent sequences
zbMATH Keywords: core sequence for double sequences
zbMATH Keywords: RH-regular matrix
reviewed by: Billy E. Rhoades
MaRDI profile type: Publication
OpenAlex ID: W1973230193
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description: scientific article; zbMATH DE number 1417977