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

Ferenc Móricz¹

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

Studia Mathematica, Tome 219 (2013) no. 2, pp. 109-121

Résumé

Let $s: [1, \infty) \to \mathbb C$ be a locally Lebesgue integrable function. We say that $s$ is summable $(L, 1)$ if there exists some $A\in \mathbb C$ such that \begin{equation} \lim_{t\to \infty} \tau(t) = A, \quad {\rm where} \quad \tau(t):= {1\over \log t} \int^t_1 {s(u) \over u}\, du.\tag{$*$}\end{equation} It is clear that if the ordinary limit $s(t) \to A$ exists, then also $\tau(t) \to A$ as $t\to \infty$. 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 $\varepsilon>0$ there exist $t_0 = t_0 (\varepsilon) > 1$ and $\lambda=\lambda(\varepsilon) > 1$ such that $$ |s(u) - s(t)| \le \varepsilon \quad {\rm whenever}\quad t_0 \le t u \le t^\lambda, $$ then the converse implication holds true: the ordinary convergence $\lim_{t\to \infty} s(t) = A$ 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

DOI : 10.4064/sm219-2-2

Keywords: infty mathbb locally lebesgue integrable function say summable there exists mathbb begin equation lim infty tau quad where quad tau log int tag * end equation clear ordinary limit exists tau infty present sufficient conditions which necessary order converse implication corollaries obtain so called tauberian theorems which analogous those known summability example function slowly oscillating which mean every varepsilon there exist varepsilon lambda lambda varepsilon varepsilon quad whenever quad lambda converse implication holds ordinary convergence lim infty follows nbsp * present necessary sufficient tauberian conditions under which ordinary convergence numerical sequence follows its logarithmic summability furthermore transparent proof earlier tauberian theorem due kwee