Nonexistence of universally accelerating linear summability methods
type: Item
instance of: scholarly article
title: Nonexistence of universally accelerating linear summability methods
zbMATH Open document ID: 0819.65006
DOI: 10.1016/0377-0427(94)90059-0
author: I. Kornfeld
published in: Journal of Computational and Applied Mathematics
publication date: 1995-09-06T00:00:00Z
review text: The equivalence of two Toeplitz matrices is deduced from their transformation properties with regard to convergent sequences. Let \(a(m,n)\) and \(b(m,n)\) \((m,n \geq 1)\) be the elements of two regular Toeplitz matrices \(A\) and \(B\). \({\mathcal S}\) being a representative convergent sequence, denote its members by \(S(n)\) \((n \geq 1)\) and its limit by \(L({\mathcal S})\). Set \(\sigma (A \| S,m) = \sum a(m,n) S(n)\) \((1 \leq n < \infty)\) for \(m \geq 1\) and define \(\sigma (B \| {\mathcal S}, m)\) for \(m \geq 1\) similarly. If, for every convergent sequence \({\mathcal S}\), \(| \sigma (B \| {\mathcal S},m) - L({\mathcal S}) | \leq K ({\mathcal S}) | \sigma (A \| {\mathcal S}, m) - L({\mathcal S}) |\) \((m \geq 1)\), where \(K({\mathcal S})\) depends upon \({\mathcal S}\) but not upon \(m\), then (a) for some \(p \geq 1\), \(a(m,n) = b(m,n)\) \((n \geq 1)\) for all \(m \geq p\) and (b) \(A\) and \(B\) sum the same sequences.
Mathematics Subject Classification ID: 65B05
Mathematics Subject Classification ID: 40C05
zbMATH DE Number: 721011
zbMATH Keywords: linear summability methods
zbMATH Keywords: Toeplitz matrices
zbMATH Keywords: summability equivalence
zbMATH Keywords: convergence acceleration
reviewed by: Peter Wynn
MaRDI profile type: Publication
full work available at URL: 0377-0427(94)90059-0
OpenAlex ID: W2094169677
cites work: Q5838698
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cites work: Sequence transformations and their applications
description: scientific article; zbMATH DE number 721011