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

Ferenc Móricz (2013)

Studia Mathematica

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 l i m t τ ( t ) = A , where τ ( t ) : = 1 / ( l o g t ) 1 t s ( u ) / u d u . (*) 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...

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