Almost convergence and Euler totient matrix
The \(\Phi\)-transform of complex-valued sequences is defined by \[ v_{n}=\sum_{k\mid n}\varphi( k) u_{k},\ n\in\mathbb{N}, \] where \(\varphi\) denotes the Euler totient function. Let \[ \widehat{c}\left(\Phi\right) :=\left\{(u_{k})\biggm\vert\exists\alpha\in\mathbb{C}:\lim_{m\rightarrow\infty}\sum_{j=0}^{m}\frac{v_{n+j}}{m+1}=\alpha\text{ uniformly in }n\right\}. \] From the authors' abstract: ``It is proved that the space \(\widehat{c}\left(\Phi\right)\) and the space of all almost convergent sequences are linearly isomorphic. Further, the \(\beta\)-dual of the space \(\widehat{c}\left(\Phi\right)\) is determined and Euler totient core of a complex-valued sequence has been defined. Finally, inclusion theorems related to this new type of core are obtained.''
scientific article; zbMATH DE number 7222413
scientific article