Schmidt’s classical Tauberian theorem says that if a sequence 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 ℝ ₊. We prove another Tauberian theorem in the case of complex-valued functions that are slowly oscillating on 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm107-1-6,
author = {Ferenc M\'oricz},
title = {Statistical extensions of some classical Tauberian theorems in nondiscrete setting},
journal = {Colloquium Mathematicae},
volume = {107},
year = {2007},
pages = {45-56},
zbl = {1112.40004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm107-1-6}
}
Ferenc Móricz. Statistical extensions of some classical Tauberian theorems in nondiscrete setting. Colloquium Mathematicae, Tome 107 (2007) pp. 45-56. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm107-1-6/