A new look at some Tauberian classes of sequences (Q1970249)

Consider some summability method \(V\). Then in general, convergence (boundedness) in the mean \(V\) does imply convergence (boundedness) of the sequence itself only if an additional assumption on the sequence, a so-called Tauberian condition (TC), is satisfied. Based on some growth function \(0<\phi(n) \uparrow\infty\) with \(\Delta \phi(n): =\phi(n)-\phi (n-1)\to 0\) two classical Tauberian conditions for a sequence \((s_n)\) are as follows: \(\Delta s_n=O (\Delta\phi (n))\) (local TC) and \[ \lim_{ \varepsilon \to 0+} \limsup_{n\to \infty} \max_{m: \phi(n) \leq\phi (m) \leq \phi(n)+ \varepsilon} |s_{m+1}- s_n|= 0\text{ (oscillation TC)}. \] The author studies the following third type of TC: For any \(\delta >0\) there exists a sequence \((t_n)\) such that \((t_n)\) satisfies the local TC and \(|t_n-s_n|<\delta\) for all \(n\). The main results of this paper study this condition and compare it with the classical ones. Among other results, it is shown that this TC is equivalent with the oscillation TC provided that \(\Delta \phi(n+1)= O(\Delta \phi(n))\). Various classical Tauberian theorems are discussed under the point of view of his results.