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Statistical extensions of some classical Tauberian theorems in nondiscrete setting. (English) Zbl 1112.40004

Summary: Schmidt’s classical Tauberian theorem says that if a sequence \((s_k : k=0,1,\ldots)\) 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.

MSC:

40E05 Tauberian theorems
40G05 Cesàro, Euler, Nörlund and Hausdorff methods
40C10 Integral methods for summability
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