A Tauberian theorem related to almost convergence (Q1192455)

Define \(A_ n^ \alpha=(\alpha+1)\ldots(\alpha+n)/n!\) and \(a_{nk}=A_{n-k}^{\alpha-1}/A_ n^ \alpha\). For a sequence \((s_ k)\) in any Banach space the summability method \(f_ A\) is defined by \(s_ k\to s(f_ A)\) if uniformly in \(p\geq 0\), \(\sum^ n_{k=0}a_{nk}s_{k+p}\to s\). For \(\alpha=1\) this convergence is known as almost convergence. For \(p=0\) one obtains \((C,\alpha)\)- convergence. The author shows that norm boundedness of the power sequence \((x^ k)\) is a Tauberian condition for its \((C,\alpha)\) summability to imply its \(f_{(C,\alpha)}\) summability.