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

• Autorzy: Ferenc Móricz
• Języki publikacji: 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.

Kategorie tematyczne

• 40E05: Tauberian theorems, general
• 40G05: Ces\`aro, Euler, N\"orlund and Hausdorff methods

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2004
• Tom: 99
• Numer: 2
• Strony: 207-219

Daty

wydano

2004

Twórcy

autor

Ferenc Móricz

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

Identyfikatory

DOI

10.4064/cm99-2-6