## Taiwanese Journal of Mathematics

### 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.

#### Article information

Source
Taiwanese J. Math., Volume 15, Number 1 (2011), 181-199.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406169

Digital Object Identifier
doi:10.11650/twjm/1500406169

Mathematical Reviews number (MathSciNet)
MR2780279

Zentralblatt MATH identifier
1238.40005

#### Citation

Chen, Chang-Pao; Chang, Chi-Tung. Tauberian Theorems for the Weighted Means of Measurable Functions of Several Variables. Taiwanese J. Math. 15 (2011), no. 1, 181--199. doi:10.11650/twjm/1500406169. https://projecteuclid.org/euclid.twjm/1500406169

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