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2013 | 219 | 2 | 109-121
Tytuł artykułu

Necessary and sufficient Tauberian conditions for the logarithmic summability of functions and sequences

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Let s: [1,∞) → ℂ be a locally Lebesgue integrable function. We say that s is summable (L,1) if there exists some A ∈ ℂ such that
$lim_{t→∞} τ(t) = A$, where $τ(t):= 1/(log t) ∫_{1}^{t} s(u)/u du$. (*)
It is clear that if the ordinary limit s(t) → A exists, then also τ(t) → A as t → ∞. We present sufficient conditions, which are also necessary, in order that the converse implication hold true. As corollaries, we obtain so-called Tauberian theorems which are analogous to those known in the case of summability (C,1). For example, if the function s is slowly oscillating, by which we mean that for every ε > 0 there exist t₀ = t₀(ε) > 1 and λ = λ(ε) > 1 such that
|s(u) - s(t)| ≤ ε whenever $t₀ ≤ t < u ≤ t^{λ}$,
then the converse implication holds true: the ordinary convergence $lim_{t→∞} s(t) = A$ follows from (*).
We also present necessary and sufficient Tauberian conditions under which the ordinary convergence of a numerical sequence $(s_{k})$ follows from its logarithmic summability. Furthermore, we give a more transparent proof of an earlier Tauberian theorem due to Kwee.
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  • Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm219-2-2
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