This paper addresses conditions for the Abel method of limitability to imply convergence and subsequential convergence.

Given ⨍ ∈ $L_l^{1}oc\left({ℝ}_{+}^2\right)$, denote by s(w,z) its integral over the rectangle [0,w]× [0,z] and by σ(u,v) its (C,1,1) mean, that is, the average value of s(w,z) over [0,u] × [0,v], where u,v,w,z>0. Our permanent assumption is that (*) σ(u,v) → A as u,v → ∞, where A is a finite number. First, we consider real-valued functions ⨍ and give one-sided Tauberian conditions which are necessary and sufficient in order that the convergence (**) s(u,v) → A as u,v → ∞ follow from (*). Corollaries allow these Tauberian conditions...

$(s_{jk}:j,k=0,1,...)$ be a double sequence of real numbers which is summable (C,1,1) to a finite limit. We give necessary and sufficient conditions under which $\left(s_{jk}\right)$ converges in Pringsheim’s sense. These conditions are satisfied if $\left(s_{jk}\right)$ is slowly decreasing in certain senses defined in this paper. Among other things we deduce the following Tauberian theorem of Landau and Hardy type: If $\left(s_{jk}\right)$ is summable (C,1,1) to a finite limit and there exist constants $n_1>0$ and H such that $jk(s_{jk}-s_{j-1,k}-s_{j-1,k}+s_{j-1,k-1})\ge -H$, $j(s_{jk}-s_{j-1,k})\ge -H$ and $k(s_{jk}-s_{j,k-1})\ge -H$ whenever $j,k>n_1$, then $\left(s_{jk}\right)$ converges. We always mean...

In this paper, we prove that a bounded double sequence of fuzzy numbers which is statistically convergent is also statistically (C, 1, 1) summable to the same number. We construct an example that the converse of this statement is not true in general. We obtain that the statistically (C, 1, 1) summable double sequence of fuzzy numbers is convergent and statistically convergent to the same number under the slowly oscillating and statistically slowly oscillating conditions in certain senses, respectively....

Let ${𝒯}_X$ denote the operator-norm closure of the class of convolution operators ${\Phi }_{\mu }:X\to X$ where $X$ is a suitable function space on $R$. Let ${ℳ}_r^p$ be the closed subspace of regular functions in the Marinkiewicz space ${ℳ}^p$, $1\le p\<\infty$. We show that the space ${𝒯}_{{ℳ}_r^p}$ is isometrically isomorphic to ${𝒯}_{L^p}$ and that strong operator sequential convergence and norm convergence in ${𝒯}_{{ℳ}_r^p}$ coincide. We also obtain some results concerning convolution operators under the Wiener transformation. These are to improve a Tauberian theorem of Wiener on ${ℳ}^2$.