On absolute summability. (English) Zbl 0649.40009

Let \(\sum^{\infty}_{n=0}x_ n\) be an infinite series with partial sums \(s_ n\). The series \(\sum^{\infty}_{n=0}x_ n\) is called absolutely summable (C,k) with index p, or simple summable \(| C,k| p\), \(p\geq 1\), if \(\sum^{\infty}_{n=1}n^{-1}| \tau^{k}_{n}| \quad p<\infty,\tau^{k}_{n}\) being the nth Cesàro mean of order k \((k>-1)\) of the sequence \((nx_ n)\). Let \(A=(a_{nk})\) be a normal matrix i.e. lower semi matrix with non-zero diagonal entries. By \((T_ n)\) we denote the A-transform of a sequence \((s_ n)\), where \(T_ n=\sum^{\infty}_{v=0}a_{nv}s_ v=\sum^{n}_{v=0}a_{nv}s_ v.\) A series \(\sum^{\infty}_{n=0}x_ n\) is called summable \(| A| p\), \(p\geq 1\), if \(\sum^{\infty}_{n=1}n^{p-1}\cdot | T_ n-T_{n- 1}| \quad p<\infty.\) The author establishes a relation between the summability methods \(| C,1| p\) and \(| A| p\), \(p\geq 1\), details of which are cited in the paper.
Reviewer: B.P.Mishra


40F05 Absolute and strong summability