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

• Autorzy: Ferenc Móricz
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 40E05: Tauberian theorems, general
• 40C10: Integral methods
• 40G99: None of the above, but in this section
• 40A05: Convergence and divergence of series and sequences
• 40A10: Convergence and divergence of integrals

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2013
• Tom: 219
• Numer: 2
• Strony: 109-121

Daty

wydano

2013

Twórcy

autor

Ferenc Móricz

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

Identyfikatory

DOI

10.4064/sm219-2-2