Ordinary convergence follows from statistical summability (C,1) in the case of slowly decreasing or oscillating sequences

Ferenc Móricz. "Ordinary convergence follows from statistical summability (C,1) in the case of slowly decreasing or oscillating sequences." Colloquium Mathematicae 99.2 (2004): 207-219. <http://eudml.org/doc/284645>.

@article{FerencMóricz2004,
abstract = {Schmidt’s Tauberian theorem says that if a sequence (xk) of real numbers is slowly decreasing and $lim_\{n→ ∞\} (1/n) ∑^\{n\}_\{k=1\} x_k = L$, then $lim_\{k→ ∞\} x_k = L$. The notion of slow decrease includes Hardy’s two-sided as well as Landau’s one-sided Tauberian conditions as special cases. We show that ordinary summability (C,1) can be replaced by the weaker assumption of statistical summability (C,1) in Schmidt’s theorem. Two recent theorems of Fridy and Khan are also corollaries of our Theorems 1 and 2. In the Appendix, we present a new proof of Vijayaraghavan’s lemma under less restrictive conditions, which may be useful in other contexts.},
author = {Ferenc Móricz},
journal = {Colloquium Mathematicae},
keywords = {statistical convergence; statistical summability ; 1); Hardy's two-sided Tauberian condition; Landau's one-sided Tauberian condition; slow decrease; slow oscillation; Vijayaraghavan's lemma},
language = {eng},
number = {2},
pages = {207-219},
title = {Ordinary convergence follows from statistical summability (C,1) in the case of slowly decreasing or oscillating sequences},
url = {http://eudml.org/doc/284645},
volume = {99},
year = {2004},
}

TY - JOUR
AU - Ferenc Móricz
TI - Ordinary convergence follows from statistical summability (C,1) in the case of slowly decreasing or oscillating sequences
JO - Colloquium Mathematicae
PY - 2004
VL - 99
IS - 2
SP - 207
EP - 219
AB - Schmidt’s Tauberian theorem says that if a sequence (xk) of real numbers is slowly decreasing and $lim_{n→ ∞} (1/n) ∑^{n}_{k=1} x_k = L$, then $lim_{k→ ∞} x_k = L$. The notion of slow decrease includes Hardy’s two-sided as well as Landau’s one-sided Tauberian conditions as special cases. We show that ordinary summability (C,1) can be replaced by the weaker assumption of statistical summability (C,1) in Schmidt’s theorem. Two recent theorems of Fridy and Khan are also corollaries of our Theorems 1 and 2. In the Appendix, we present a new proof of Vijayaraghavan’s lemma under less restrictive conditions, which may be useful in other contexts.
LA - eng
KW - statistical convergence; statistical summability ; 1); Hardy's two-sided Tauberian condition; Landau's one-sided Tauberian condition; slow decrease; slow oscillation; Vijayaraghavan's lemma
UR - http://eudml.org/doc/284645
ER -