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

Let $s: [1, \infty) \to \C$ be a locally integrable function in Lebesgue's sense on the infinite interval $[1, \infty)$. We say that $s$ is summable $(L, 1)$ if there exists some $A\in \C$ such that $$\lim_{t\to \infty} \tau(t) = A, \quad {\rm where} \quad \tau(t):= {1\over \log t} \int^t_1 {s(u) \over u} du.\leqno(*)$$ It is clear that if the ordinary limit $s(t) \to A$ exists, then the limit $\tau(t) \to A$ also exists 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 $\e>0$ there exist $t_0 = t_0 (\e) > 1$ and $\lambda=\lambda(\e) > 1$ such that $$|s(u) - s(t)| \le \e \quad {\rm whenever}\quad t_0 \le t < u \le t^\lambda,$$ 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. Among others, we give a more transparent proof of an earlier Tauberian theorem due to Kwee [3].