Statistical extension of classical Tauberian theorems in the case of logarithmic summability of locally integrable functions on $[1,\infty)$

Abstract

Let $s:[1,\infty) \to \C $ be a locally integrable function in Lebesgue's sense. The logarithmic (also called harmonic) mean of the function $s$ is defined by [\tau(t) := \frac 1{\log t} \int_1^t \frac {s(x)}{x} dx, \qquad t>1,] where the logarithm is to base $e$. Besides the ordinary limit $\lim_{x\to \infty} s(x)$, we also use the notion of the so-called statistical limit of $s$ at $\infty$, in notation: $ \stlim_{x\to \infty} s(x)=\ell $, by which we mean that for every $\e>0$, [\lim_{b\to \infty} \frac 1b \Big | \Big {x\in(1,b): |s(x)-\ell| >\e \Big} \Big| = 0.] We also use the ordinary limit $\lim_{t\to\infty} \tau(t)$ as well as the statistical limit $\stlim_{t\to\infty} \tau(t)$. We will prove the following Tauberian theorem: Suppose that the real-valued function $s$ is slowly decreasing or the complex-valued $s$ is slowly oscillating. If the statistical limit $\stlim_{t\to\infty} \tau(t) =\ell $ exists, then the ordinary limit $\lim_{x\to\infty} s(x) = \ell $ also exists.