Statistical extensions of some classical Tauberian theorems in nondiscrete setting

Tom 107 / 2007

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


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.


  • Ferenc M/oriczBolyai Institute
    University of Szeged
    Aradi vértanúk tere 1
    6720 Szeged, Hungary

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