Abel extensions of some classical Tauberian theorems

Gül, Erdal and Albayrak, Mehmet

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creative_2019_28_2_105_112

DOI: https://doi.org/10.37193/CMI.2019.02.02

Issue no: Vol 28/2019 no. 2 Tags: discrete Abel convergence, Tauberian theorems, Slowly oscillating, Abel convergence

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The well-known classical Tauberian theorems given for $A_{\lambda}$ (the discrete Abel mean) by Armitage and Maddox in [Armitage, H. D and Maddox, J. I., Discrete Abel means, Analysis, 10 (1990), 177–186] is generalized. Similarly the “one-sided” Tauberian theorems of Landau and Schmidt for the Abel method are extended by replacing $\lim As$ with Abel- $\lim A\sigma_{n}^{i}(s).$
Slowly oscillating of $\{s_{n}\}$ is a Tauberian condition of the Hardy-Littlewood Tauberian theorem for Borel summability which is also given by replacing $\lim_{t} (Bs)_{t} = \ell$ , where t is a continuous parameter,
with $\lim_{n}(Bs)_{n} = \ell$ , and further replacing it by Abel- $\lim (B\sigma_{k}^{i}(s))_{n} = \ell$ , where B is the Borel matrix method.