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

Ferenc Móricz

Studia Mathematica, Tome 215 (2013), p. 109-121 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2013-01-01

Zbl 1292.40003

EUDML-ID : urn:eudml:doc:286286