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# Matrix transformations based on Dirichlet convolution

Published:1997-12-01
Printed: Dec 1997
• Chikkanna Selvaraj
• Suguna Selvaraj
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## Abstract

This paper is a study of summability methods that are based on Dirichlet convolution. If $f(n)$ is a function on positive integers and $x$ is a sequence such that $\lim_{n\to \infty} \sum_{k\le n} {1\over k}(f\ast x)(k) =L$, then $x$ is said to be {\it $A_f$-summable\/} to $L$. The necessary and sufficient condition for the matrix $A_f$ to preserve bounded variation of sequences is established. Also, the matrix $A_f$ is investigated as $\ell - \ell$ and $G-G$ mappings. The strength of the $A_f$-matrix is also discussed.
 MSC Classifications: 11A25 - Arithmetic functions; related numbers; inversion formulas 40A05 - Convergence and divergence of series and sequences 40C05 - Matrix methods 40D05 - General theorems