Arithmetical summability
type: Item
instance of: scholarly article
title: Arithmetical summability
zbMATH Open document ID: 1256.40001
author: William H. Ruckle
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-10-19T00:00:00Z
review text: A scalar sequence \(f=(f(n))\) is called arithmetically summable if for each \(\varepsilon >0\) there exists \(n\in\mathbb{N}\) such that \(\left| \sum_{k\mid m}f( k)-\sum_{k\mid \left\langle m,n\right\rangle}f( k) \right|<\varepsilon \) for every \(m\in \mathbb{N}\). (Here \(\sum_{k\mid m}\) stands for the finite sum over the positive integers \(k\) that divide \(m\) and \(\langle m,n\rangle\) is the greatest common divisor of \(m\) and \(n\).) A sequence \(g\) is called arithmetically convergent if for each \(\varepsilon >0\) there exists \(n\in \mathbb{N}\) such that \(\left| g(m) -g(\langle m,n\rangle) \right|<\varepsilon \) for every \(m\in \mathbb{N}\). The sets of arithmetically summable and arithmetically convergent sequences are denoted by \(\mathbf{AS}\) and \(\mathbf{AC}\), respectively. Obviously, \(f\in \mathbf{AS}\) if and only if \(g:=\mathbf{W}f\in \mathbf{AC}\), where \(\mathbf{W}=(w_{nk})\) is the lower triangular matrix with \(w_{nk}=1\) if \(k\mid n\) and \(w_{nk}=0\) otherwise. The author shows that \(\mathbf{AC}\) is a proper subalgebra of \(\mathbf{QP}\), the algebra of semiperiodic sequences defined by \textit{I. D. Berg} and \textit{A. Wilansky} [Mich. Math. J. 9, 363--368 (1962; Zbl 0128.34001)]. Moreover, \(\mathbf{AC}\) is isomorphic to \(C[0,1]\). The space \(\mathbf{AS}\) is the closure of \(\varphi\), the space of all finite sequences, in the BK-space \[ \mathbf{BAS}:=\left\{ f=(f(k))\mid| f| :=\sup_{n}\left| \sum_{k\mid m}f(k)\right| <\infty \right\} . \] It is proved that (a) \(\mathbf{AS}^{\ast}\), the dual sequence space (or \(f\)-dual) of \(\mathbf{AS}\), is the multiplier algebra \(\mathbf{BAS}\rightarrow \mathbf{BAS}\), i.e., \(f\in \mathbf{AS}^{\ast}\) if and only if \(fg:=(f(k)g(k))\in \mathbf{BAS}\) for every \(g\in \mathbf{BAS}\), (b) \(\mathbf{AS}_{0}^{\ast}\), the closure of \(\varphi\) in the BK-space \(\mathbf{AS}^{\ast}\), has the form \(\mathbf{W}^{\text{T}}(\ell ^{1})\), where \(\mathbf{W}^{\text{T}}\) is the transpos…
reviewed by: Toivo Leiger
Mathematics Subject Classification ID: 40A05
Mathematics Subject Classification ID: 46A45
zbMATH DE Number: 6096069
zbMATH Keywords: sequence space
zbMATH Keywords: arithmetical sum
zbMATH Keywords: Möbius inversion
zbMATH Keywords: summability
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.06.048
OpenAlex ID: W4206668654
cites work: Topologies on sequence spaces
cites work: Q3259107
cites work: Determinant and inverse of meet and join matrices
cites work: On Smith's determinant
cites work: The algebra of semiperiodic sequences
cites work: Periodic, almost-periodic, and semiperiodic sequences
cites work: The conjugate space of the space of semiperiodic sequences
cites work: Multipliers on the Space of Semiperiodic Sequences
cites work: Q3530847
cites work: Q4293080
cites work: The discrete Fourier transform of $r$-even functions
cites work: Infinite matrices summing every almost periodic sequence
cites work: Q5667074
cites work: Q3254057
DOI: 10.1016/J.JMAA.2012.06.048
Recommended article: Q4848401
Recommended article: Q5654097
Recommended article: Arithmetical progressions and the number of sums
Recommended article: Arithmetic progressions in sumsets
Recommended article: On arithmetic properties of sumsets
Recommended article: Finite sums of arithmetic progressions
Recommended article: Q5314785
Recommended article: Q3481793
description: scientific article; zbMATH DE number 6096069