General Nörlund transforms and power series methods
Together with general power series methods \((P)\) generated by positive sequences \((p_ n)\) the general Nörlund transforms \((N,p^{*\alpha},p)\), \(\alpha \in \mathbb{N}\) are considered. Here \[ p^{*1}_ n=p_ n,\;p_ n^{*(\alpha+1)}=\sum^ n_{k=0} p^{*\alpha}_{n-k} p_ k \] and we say for a sequence \((s_ n)\), that \(s_ n \to s(N,p^{*\alpha},p)\), if \[ {1 \over p^{*(\alpha+1)}_ n} \sum^ n_{k=0} p^{*\alpha}_{n-k} p_ ks_ k \to s,\;(n \to \infty). \] These methods generalize the well known pairs Cesàro-Abel and Euler-Borel. The inclusions \((N,p^{*\alpha},p) \subseteq (N,p^{*\beta},p) \subseteq (P)\), \(\alpha \leq \beta\), \(\alpha,\beta \in \mathbb{N}\) are established, as well as the tauberian implication under the condition \(s_ n={\mathbf O} (1)\). Furthermore Cesàro convergence with speed is related to \((N,p^{*\alpha},p)\)-convergence generalizing the well known Cesàro to Euler case.
scientific article; zbMATH DE number 567441
scientific article