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

Ferenc Móricz¹

¹ Bolyai Institute University of Szeged Aradi vértanúk tere 1 6720 Szeged, Hungary

Colloquium Mathematicum, Tome 99 (2004) no. 2, pp. 207-219

Résumé

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.

Détail

DOI : 10.4064/cm99-2-6

Keywords: schmidts tauberian theorem says sequence real numbers slowly decreasing mathop lim infty sum mathop lim infty notion slow decrease includes hardys two sided landaus one sided tauberian conditions special cases ordinary summability replaced weaker assumption statistical summability schmidts theorem recent theorems fridy khan corollaries theorems appendix present proof vijayaraghavans lemma under restrictive conditions which may useful other contexts