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• # Artykuł - szczegóły

## Colloquium Mathematicum

2004 | 99 | 2 | 207-219

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

EN

### Abstrakty

EN
Schmidt's Tauberian theorem says that if a sequence (x_k) 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.

207-219

wydano
2004

### Twórcy

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