We prove ratio Tauberian theorems for relatively bounded functions and sequences in Banach spaces.

This article is a survey of some Tauberian theorems obtained recently in connection with work on asymptotic behaviour of semigroups of operators on Banach spaces. The results in operator theory are described in [6], where we made little attempt to show the Tauberian aspects. At the end of this article, we will give a sketch of the connections between the results in this article and in [6]; for details, the reader can turn to the original papers. In this article, we make no attempt to describe...

Schmidt’s classical Tauberian theorem says that if a sequence $(s_k:k=0,1,...)$ of real numbers is summable (C,1) to a finite limit and slowly decreasing, then it converges to the same limit. In this paper, we prove a nondiscrete version of Schmidt’s theorem in the setting of statistical summability (C,1) of real-valued functions that are slowly decreasing on ℝ ₊. We prove another Tauberian theorem in the case of complex-valued functions that are slowly oscillating on ℝ ₊. In the proofs we make use of two nondiscrete...

We give a sufficient condition for a non-negative random variable $X$ to be of Pareto type by investigating the Laplace-Stieltjes transform of the cumulative distribution function. We focus on the relation between the singularity at the real point of the axis of convergence and the asymptotic decay of the tail probability. For the proof of our theorems, we apply Graham-Vaaler’s complex Tauberian theorem. As an application of our theorems, we consider the asymptotic decay of the stationary distribution...

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...