Tauberian Theorems for the Weighted Means of Measurable Functions of Several Variables

Abstract

Let $f, \omega : \mathbb{R}_+^n \to \mathbb{C}$ and $T_{\omega} f(x)$ denote the weighted mean of $f$ at $x$ with respect to the weight function $\omega$. We prove that the conditions of slow oscillation and slow decrease are Tauberian conditions for the implications: $f(x) \stackrel{st}{\rightarrow} l \Longrightarrow f(x) \rightarrow l$ and $T_{\omega} f(x) \stackrel{st}{\rightarrow} l \Longrightarrow f(x) \rightarrow l$. We also prove that the statistical version of the conditions of deferred means are Tauberian conditions for the implication: $T_{\omega} f(x) \stackrel{st}{\rightarrow} l \Longrightarrow f(x) \stackrel{st}{\rightarrow} l$. These generalize several well-known results.