Statistical extensions of some classical Tauberian theorems in nondiscrete setting

Ferenc M/oricz¹

¹ Bolyai Institute University of Szeged Aradi vértanúk tere 1 6720 Szeged, Hungary

Colloquium Mathematicum, Tome 107 (2007) no. 1, pp. 45-56

Résumé

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.

Détail

DOI : 10.4064/cm107-1-6

Keywords: schmidts classical tauberian theorem says sequence mathinner ldotp ldotp ldotp real numbers summable finite limit slowly decreasing converges limit paper prove nondiscrete version schmidts theorem setting statistical summability real valued functions slowly decreasing mathbb prove another tauberian theorem complex valued functions slowly oscillating mathbb proofs make nondiscrete analogues famous vijayaraghavan lemma which seem may useful other contexts