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

TL;DR

This paper establishes necessary and sufficient Tauberian conditions for the logarithmic summability of functions and sequences, providing new insights and simpler proofs for classical results in summability theory.

Contribution

It introduces precise Tauberian conditions linking logarithmic summability to ordinary convergence for functions and sequences, including a clearer proof of Kwee's theorem.

Findings

Characterization of when logarithmic summability implies convergence

Necessary and sufficient Tauberian conditions for functions and sequences

Simplified proof of Kwee's Tauberian theorem

Abstract

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].