In this paper we give the form of every multiplicative Hausdorff prime matrix, thus answering a long-standing open question.

The paper offers a study of the inverse Laplace transforms of the functions $I_n(rs)\{sI_n^{'}(s)\}^{-1}$ where $I_n$ is the modified Bessel function of the first kind and $r$ is a parameter. The present study is a continuation of the author's previous work %[\textit{Canadian Mathematical Bulletin} 45] on the singular behavior of the special case of the functions in question, $r$=1. The general case of $r \in [0,1]$ is addressed, and it is shown that the inverse Laplace transforms for such $r$ exhibit significantly more complex behavior than their predecessors, even though they still only have two different types of points of discontinuity: singularities and finite discontinuities. The functions studied originate from non-stationary fluid-structure interaction, and as such are of interest to researchers working in the area.

The main results deal with conditions for the validity of the weighted convolution inequality $\sum_{n\in\mathbb Z}\left|b_n\sum_{k\in\mathbb Z} a_{n-k}x_k\right|^p\le C^p\sum_{k\in\mathbb Z} |x_k|^p$ when $p\ge1$.

The weighted mean matrix $M_a$ is the triangular matrix $\{a_k/A_n\}$, where $a_n > 0$ and $A_n := a_1 + a_2 + \cdots + a_n$. It is proved that, subject to $n^c a_n$ being eventually monotonic for each constant $c$ and to the existence of $\alpha := \lim \frac{A_n}{na_n}$, $M_a \in B(l_p)$ for $1 < p < \infty$ if and only if $\alpha < p$.

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.

The two theorems proved yield simple yet reasonably general conditions for triangular matrices to be bounded operators on $l_p$. The theorems are applied to N\"orlund and weighted mean matrices.

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.