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

TL;DR

This paper extends classical Tauberian theorems to the case of logarithmic summability for locally integrable functions, establishing conditions under which statistical limits imply ordinary limits.

Contribution

It introduces a new Tauberian theorem linking statistical limits of logarithmic means to ordinary limits for functions with specific oscillation or decrease properties.

Findings

Statistical limit of the logarithmic mean implies the ordinary limit.

The theorem applies to slowly decreasing or oscillating functions.

Provides a bridge between statistical and classical convergence notions.

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.