In this paper we prove the convergence of arithmetic means of sequences generated by a periodic function $\varphi (x) $, moreover if $\varphi (x) $ satisfies a suitable symmetry condition, we prove that their limit is $\varphi (0) $. Applications of previous results are given to study other means of sequences and the behaviour of a class of recursive series.

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.