Móricz, Ferenc Statistical extensions of some classical Tauberian theorems in nondiscrete setting. (English) Zbl 1112.40004 Colloq. Math. 107, No. 1, 45-56 (2007). 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. Cited in 3 Documents MSC: 40E05 Tauberian theorems 40G05 Cesàro, Euler, Nörlund and Hausdorff methods 40C10 Integral methods for summability Keywords:statistical limit of measurable functions at \(\infty\); statistical summability \((C; 1)\) of locally integrable functions on \({\mathbb R}_+\); slow decrease; slow oscillation; Landau’s one-sided Tauberian condition; Hardy’s two-sided Tauberian condition; nondiscrete analogues of Vijayaraghavan’s lemma; convergence of improper integrals PDFBibTeX XMLCite \textit{F. Móricz}, Colloq. Math. 107, No. 1, 45--56 (2007; Zbl 1112.40004) Full Text: DOI