Arithmetical summability
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 transpose matrix of \(\mathbf{W}\).
scientific article; zbMATH DE number 6096069
scientific article