Nörlund and weighted mean matrices as operators on \(\ell_ p\)
type: Item
instance of: scholarly article
title: Nörlund and weighted mean matrices as operators on \(\ell_ p\)
zbMATH Open document ID: 0712.47023
published in: Rocky Mountain Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: Let f be a logarithmic-exponential function defined on \([x_ 0,\infty)\) for some \(x_ 0>n\), and let \(\{a_ n\}\) be a sequence of positive numbers with \(a_ n=f(n)\) for sufficiently large n. Write \(A_ n=\sum^{n}_{k=0}a_ k\). Define the Nörlund matrix \(N_ a=(a_{nk})\) by \[ a_{nk}=a_{n-k}/A_ n\text{ for } 0\leq k\leq n,\text{ and } =0\text{ for } k>n, \] and the weighted mean matrix \(W_ a=(a_{nk})\) by \[ a_{nk}=a_ k/A_ n\text{ for } 0\leq k\leq n,\text{ and } =0\text{ for } k>n. \] For \(1<p<\infty\), let \(\ell_ p\) be the BK- space of all sequences \(\{x_ k\}\) such that \(\sum | x_ k|^ p\) converges. Denote by \(B(\ell_ p)\) the Banach algebra of all bounded linear operators on \(\ell_ p\) with the norm \(\| T\|_ p\) for T in \(B(\ell_ p)\). The authors have established that (i) \(Na\in B(\ell_ p)\Leftrightarrow \lim_{n\to \infty}(na_ n/A_ n)<\infty;\) (ii) \(W_ a\in B(\ell_ p)\Leftrightarrow \lim_{n\to \infty}(A_ n/na_ n)<p.\) Asymptotic properties of the norms of the operators are also obtained.
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 40G05
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 26A12
zbMATH DE Number: 4171754
zbMATH Keywords: logarithmic-exponential function
zbMATH Keywords: Nörlund matrix
zbMATH Keywords: weighted mean matrix
zbMATH Keywords: BK-space
zbMATH Keywords: Asymptotic properties of the norms of the operators
author: Frank P. Cass
author: Werner Kratz
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181073159
OpenAlex ID: W2003189900
cites work: Nörlund matrices as bounded operators on \(\ell_ p\)
cites work: Matrix operators on \(\ell^p\)
cites work: The Tauberian theorems which interrelate different Riesz means
cites work: SOME ELEMENTARY INEQUALITIES, III
DOI: 10.1216/RMJM/1181073159
description: scientific article; zbMATH DE number 4171754