Statistical extensions of some classical Tauberian theorems in nondiscrete setting

Ferenc Móricz. "Statistical extensions of some classical Tauberian theorems in nondiscrete setting." Colloquium Mathematicae 107.1 (2007): 45-56. <http://eudml.org/doc/284150>.

@article{FerencMóricz2007,
abstract = {Schmidt’s classical Tauberian theorem says that if a sequence $(s_\{k\}: k = 0,1,...)$ 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.},
author = {Ferenc Móricz},
journal = {Colloquium Mathematicae},
keywords = {statistical limit of measurable functions at ; statistical summability ; 1)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},
language = {eng},
number = {1},
pages = {45-56},
title = {Statistical extensions of some classical Tauberian theorems in nondiscrete setting},
url = {http://eudml.org/doc/284150},
volume = {107},
year = {2007},
}

TY - JOUR
AU - Ferenc Móricz
TI - Statistical extensions of some classical Tauberian theorems in nondiscrete setting
JO - Colloquium Mathematicae
PY - 2007
VL - 107
IS - 1
SP - 45
EP - 56
AB - Schmidt’s classical Tauberian theorem says that if a sequence $(s_{k}: k = 0,1,...)$ 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.
LA - eng
KW - statistical limit of measurable functions at ; statistical summability ; 1)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
UR - http://eudml.org/doc/284150
ER -