A new type of Cesàro mean. (English) Zbl 0693.40009

In respect of \(\lambda =(\lambda_ n)\), \(0\leq \lambda_ 0<...<\lambda_ n\to \infty\), the authors define \((s_ n)\in C_{\lambda}\) by convergence of \((1+\lambda_ n)^{-1}\sum_{k\leq \lambda_ n}s_ k\) (n\(\to \infty)\), thus generalizing the method \((C,1)=C_{{\mathbb{N}}}\). Abelian results embed the inclusions \((C,1)\subseteq C_{\lambda}\) (Corollary 2, which means that \(\sigma_ n\to \ell\) infers \(\sigma_{n_ k}\to \ell\) where \(n_{k-1}\leq n_ k\to \infty)\) and (C,1)\(\subsetneqq A\) with A the Abel method. The main result is Tauberian, saying that a real sequence \((s_ n)\in C_{\lambda}\) is, in case of \(\lambda_{n+1}/\lambda_ n\to 1\), limitable (C,1) if \(s_ m-s_ n>-\epsilon\) whenever \(m>n\geq n(\epsilon)\) and (m-n)/n\(\leq \delta (\epsilon)\); under \(\lambda_{n+1}/\lambda_ n=O(1)\), the conclusion continues to hold with \(\delta\) (\(\epsilon)\) being replaced by some constant.
Reviewer: H.H.Körle


40E05 Tauberian theorems
40G10 Abel, Borel and power series methods
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