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

Ferenc Móricz. "Necessary and sufficient Tauberian conditions for the logarithmic summability of functions and sequences." Studia Mathematica 219.2 (2013): 109-121. <http://eudml.org/doc/286286>.

@article{FerencMóricz2013,
abstract = {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.},
author = {Ferenc Móricz},
journal = {Studia Mathematica},
keywords = {logarithmic summability of functions and sequences; Tauberian conditions; slow decrease and oscillation; inclusion theorems},
language = {eng},
number = {2},
pages = {109-121},
title = {Necessary and sufficient Tauberian conditions for the logarithmic summability of functions and sequences},
url = {http://eudml.org/doc/286286},
volume = {219},
year = {2013},
}

TY - JOUR
AU - Ferenc Móricz
TI - Necessary and sufficient Tauberian conditions for the logarithmic summability of functions and sequences
JO - Studia Mathematica
PY - 2013
VL - 219
IS - 2
SP - 109
EP - 121
AB - 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.
LA - eng
KW - logarithmic summability of functions and sequences; Tauberian conditions; slow decrease and oscillation; inclusion theorems
UR - http://eudml.org/doc/286286
ER -