Ordinary convergence follows from statistical summability $(C,1)$ in the case of slowly decreasing or oscillating sequences
Volume 99 / 2004
Colloquium Mathematicum 99 (2004), 207-219
MSC: Primary 40E05, 40G05.
DOI: 10.4064/cm99-2-6
Abstract
Schmidt's Tauberian theorem says that if a sequence $(x_k)$ of real numbers is slowly decreasing and $\mathop {\rm lim}_{n\to \infty } (1/n) \sum ^n_{k=1} x_k = L$, then $\mathop {\rm lim}_{k\to \infty } x_k = L$. 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.
