## Statistical extensions of some classical Tauberian theorems in nondiscrete setting

### Volume 107 / 2007

Colloquium Mathematicum 107 (2007), 45-56
MSC: Primary 40C10, 40E05, 40G05.
DOI: 10.4064/cm107-1-6

#### Abstract

Schmidt's classical Tauberian theorem says that if a sequence $(s_k : k=0,1,\mathinner {\ldotp \ldotp \ldotp })$ 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 ${{\mathbb R}}_+$. We prove another Tauberian theorem in the case of complex-valued functions that are slowly oscillating on ${{\mathbb R}}_+$. In the proofs we make use of two nondiscrete analogues of the famous Vijayaraghavan lemma, which seem to be new and may be useful in other contexts.