Necessary and sufficient Tauberian conditions for certain weighted mean methods of summability
Let \(p_0> 0\), \(p_n\geq 0\), \(P_n:= \sum_0^n p_k\to \infty\). A sequence \((s_k)\) is summable by the weighted mean \((M, p)\) [or \((\overline{N}, p)]\) if there is a finite \(s\) such that \(t_n:= P_n^{-1} \sum_0^n p_k s_k \to s\). Now suppose that \[ \begin{aligned} 1&< \liminf (P_n/ P_{[\lambda n]})\leq \limsup (P_n/ P_{[\lambda n]})< \infty \qquad (0< \lambda< 1)\\ \text{and} 1 &< \liminf (P_{[\lambda n]}/ P_n)\leq \limsup (P_{[\lambda n]}/ P_n)< \infty \qquad (\lambda>1). \end{aligned} \] It is then shown (Theorem 1) that (for real \(s_n\)) \(t_n\to s\Rightarrow s_n\to s\) if and only if \[ \begin{aligned} &\limsup_{\lambda\searrow 1} \liminf_{n\to \infty} (P_{[\lambda n]}- P_n )^{-1} \sum_{k= n+1}^{[\lambda n]} p_k (s_k- s_n) \geq 0\\ \text{and} &\limsup_{\lambda\nearrow 1} \liminf_{n\to \infty} (P_n- P_{[\lambda n]} )^{-1} \sum_{k= [\lambda n]+1}^n p_k (s_n- s_k )\geq 0. \end{aligned} \] In particular, these last conditions hold when \((s_k)\) is showly decreasing. There is an extension (Theorem 2) for complex sequences \((s_k)\), where the conditions are satisfied in particular in the classical Tauberian case \(s_k- s_{k-1}= O(k^{-1})\).
scientific article; zbMATH DE number 755495