On a translated Cesàro type summability method. I (Q2716691)

K. Ishiguro defined the \((D,\propto)\) summability method \((\propto >0)\). In this paper the author defines the \((D_\lambda)\)-summablity method which generalizes the \((D,\propto)\) method. \((D_\lambda)\)-summability is translative if, whenever \(\{S_n\}_0^\infty\) is summable \((D_\lambda)\) to \(s\), the sequence \(\{S_n^*\}_0^\infty\), where \(S_0^*=S_0\) and \(S_n^* =S_{n-1}\), \((n\geq 1)\), is summable \((D_\lambda)\) to \(s\), and conversely. The author also defines \((D_\lambda)\)-summability to \(s\) of functions of continuous variables, and he proves some theorems analogous to the theorem stated for \((D_\lambda)\)-summability.